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Certainly flustrated “is a word”, although it does not appear to be especially well thought of. The OED reports that the verb flustrate has been used for more than 300 years; it simply means fluster.

Here’s one amusing citation:

• 1876 Mrs. Oliphant Curate in Charge (ed. 5) II. iv. 100 — The head of the college was slightly flustrated, if such a vulgar word can be used of such a sublime person.

It is, however, marked vulgar or jocular — as so too is
flustration (originally sometimes spelled flusteration), which has been around nearly as long and is reported to mean:

The condition of being flustered; ‘fluster’, agitation.

I would probably avoid using flustrate and its inflected and derived forms in formal contexts unless I were trying to convey a folksy, jocular, or ironic feel, such as in reported speech. But I wouldn’t let it confusticate or bebother me, either.

It is a jocular blend of fluster and frustated.

I'd probably burninate the word if I could, but please don't misunderestimate all people who use it. Japanese people sometimes mix up "r" and "l" when they're typing as well as when they're speaking, in which case "flustrated" is just a flustrating mistake:

Properties and
phase transitions in flustrated Ising systems. In Frustrated Spin
Systems (eds. Diep, H.). World Scientific, Singapore, pp. 59–106.

(Note: the error was probably not in the article cited, but in the citation itself)

In as much as the Oxford English Dictionary defines it as "to fluster," it's generally received as a portmanteau of "fluster" and "frustrate." English is chock-full of portmanteaus: spork (spoon+fork), biopic (biographical+picture), carjack (car+hijack), motel (motor+hotel), paratrooper (parachute+trooper), and many, many others. Even the word portmanteau is a portmanteau of "porter" and "manteau," meaning mantle, and refers to a type of suitcase, which word Lewis Carol then borrowed in order to coin its now more common meaning as word that packs multiple words into a single word, like Jabberwocky does.

That said, I am not in favor of all portmanteaus. Far from it. For me, in fact, some of the most insipid and insidious words in the English language are Portmanteaus that describe people's relationships, as in Brangelina (Brad Pitt + Angelina Jolie) and Bennifer (Ben Affleck + Jennifer Garner). Despite these particular assaults leveled by pop culture against my better sanity, however, I would still not toss out the portmanteau because so many words, particularly technological words like email (electronic+mail), come into English directly as a result of portmanteaus, so let's not get too flustrated by them.

Dictionary definitions of 'flustrate' and allied words

As tchrist's answer might lead you to expect, the Compact Edition of the Oxford English Dictionary (1971) contains entries for flustrate and flustration, with citations going back to 1712 and 1748, respectively. Here are the entries, minus the citations:

Flustrate v. vulgar or jocular. Also flusterate. {f. FLUSTER v. + -ATE ["a verbal formative, used to english L. verbs of the first conjugation, and to form Eng. verbs on other L. words or elements"]} = FLUSTER v. ² ["trans. To flush or excite with drink, make half-tipsy"] and ⁴ ["trans. To flurry, confuse"]. [Citations omitted.]

Flustration n. vulgar or jocular. Also flusteration. {f. FLUSTER v. + -ATION} The condition of being flustered; 'fluster', agitation. [Citations omitted.]

These definitions don't show any intermingling of the sense of words fluster and frustrate (or frustration); instead they indicate that the meanings of flustrate are a subset of the meanings of fluster (two of the four definitions that OED identifies for the latter). And OED gives flustration essentially the same definition that it assigns to flusterment ("the state of being flustered"). Whether this accurately depicts today's usage of flustered today is a question I'll address later.

OED's earliest citations for fluster, flust[e]ration, and flusterated come from Richard Steele in The Spectator, Samuel Richardson in Clarissa, and Mrs. Agnes Bennett in Beggar Girl. The instance by Steele in The Spectator No. 493 (September 25, 1712/1883) originally read as follows (in the words of a character named "Jack Toper of the Temple"):

Thomas [a servant] that lived with me was turned away because he was too good for me. You know I live in Taverns; he is an orderly sober Rascal, and thinks much to sleep in an Entry till two in a Morning. He told me one day when he was dressing me, that he wondered I was not dead before now, since I went to Dinner in the Evening, and went to Supper at two in the Morning. We were coming down Essex-Street one Night a little flustrated, and I was giving him the Word to alarm the Watch; he had the Impudence to tell me that it was against the Law.

However, the edition of 1739 has flustred in place of flustrated (and on the same model as wondred, which replaces the original wondered earlier in the paragraph). Flustred likewise appears in the edition of 1747 and the edition of 1755. Next, as early as the edition of 1778, editors substituted flustered for flustrated in the sentence, indicating, and flustered appears in this number of The Spectator in vast majority of subsequent editions of The Spectator for the next two centuries. In the many editions of The Spectator viewable at Hathi Trust, the earliest to show flustrated in its original place is the Henry Morley edition, of 1883, which (accurately) bills itself as "reproducing the original text both as first issued and as corrected by its authors." The upshot of all this is that very few people between 1713 and 1883—except those who had read the original issue no. 493 of The Spectator in 1712—were ever exposed to Steele's use of flustrated; and the editor's alteration of it in 1739 suggests that at that time it was not a very well-known word.

Next, from Richardson's Clarissa, or, The History of a Young Lady (1748/1768):

Impertinence! said I—Wert thou bid to come up in this fluttering way?—And I took up my fan, and fanned myself.

Bless me! said she, how soon these fine young Ladies will be put into flusterations!—I meant not either to offend or frighten you, I am sure.—

And from Bennett's The Beggar Girl and Her Benefactors (1797):

Although Mrs. Betty, who had been brought up by Mrs. Feversham, retained from habit a kind of respect and regard for her, she was, she confessed, quite flusterated at the idea of her interference in those domestic arrangements of her master, which, she thought, could not be in better hands than her own; and with respect to Rosa,—who could take more care of her than she did herself?

It seems relevant that Johnson's A Dictionary of the English Language (1756) has only the drink-related meaning for the verb fluster (and no entry at all for flustrate):

To FLUSTER. v. a. {from To flush} To make hot and rosy with drinking. Shakespeare.

John Brockett, A Glossary of North Country Words, in Use (1825) has this brief entry:

FLUSTERATION, hurry, confusion, sudden impulse.

On the American side of the Atlantic, David Humphreys includes flusteration in a glossary attached to his play The Yankey in England (1815):

flustration, extreme agitation.

John Bartlett, Dictionary of Americanisms (1848) notes Brockett's entry in this entry for the same word:

FLUSTERATION. Heat; hurry; confusion.—Brockett's Glossary. A vulgar word also heard among ourselves.

J.S. Farmer & W.E. Henley, Slang and Its Analogues, volume 3 (1893) separates its coverage of flustrated into two entries—one for flustered and the other for flusticated:

FLUSTERED (or FLUSTRATED) ppl. adj. (old).—Excited by drink, circumstances, another person's impudence, etc.; also mildly drunk. [Citations omitted.]

FLUSTICATED, or FLUSTRATED, ppl. adj. (old and colloquial).—Confused; in a state of heat or excitement. [Citations omitted.]

FLUSTRATION, subs. (old and colloquial).—Heat; excitement; bustle; confusion; FLURRY (q.v.). [Citations omitted.]

Mitford Mathews, A Dictionary of Americanisms on Historical Principles (1951) has no entries for flustrated or flustration, but offers this note on flusticated:

flusticated a. {App. f. fluster, v., and intoxicated.} Flustered or fuddled, as with drink. Slang Obs. [Citations omitted.]

It is noteworthy here that Mathews considers flusticated to have the meaning that Farmer & Henley assign to "flustered (or flustrated)." Even more mysteriously, Farmer & Henley offer as their first citation for "flusticated or flustrated" (in the nontipsy sense) Steele's 1712 instance from The Spectator, in which "a little flustrated" almost certainly does have the sense "a little drunk," given that the speaker there emphasizes his fondness for drink (right down to his name, Toper).

Dictionaries in the Merriam-Webster Collegiate Dictionary series ignore flustrate and closely related terms entirely, but the big Webster's Third New International Dictionary (1961/1986) has this brief entry for flustrated:

flustrated or flusterated adj {fluster + -ated (as in frustrated)} : FLUSTERED

The reference to frustrated here is by way of similarity in the suffix. Webster's does not suggest that the core sense of frustrated (balked, thwarted, or disappointed) is even a secondary element in flustrated. The American Heritage Dictionary of the English Language, fifth edition (2011) doesn't acknowledge flustrate, flustrated, flustration, etc., as mainstream English words.

Early historical occurrences of the 'flustrate' family

Google Books searches turn up a few additional matches from the eighteenth and early nineteenth centuries. For example, from Arthur Murphy, The Upholsterer, or What News? second edition (1765):

Enter TERMAGANT.

TERMAGANT. Undone, Undone! I'm all over in a flustration—old Jimini Gomini's coming.

HARRIET. O Lud, what is to be done now?

TERMAGANT. The Devil! what can be done? I have it—don't flustrate yourself—I'll find some Nonsense News for him—away with you both into that Room. Quick, quick. {They Exit.

From a letter from Robert Callender to Governor Penn (April 21, 1771) in Pennsylvania Archives, volume 4 (1853):

I have been very Cautious in Mentioning any thing of this, as it woud Allarme all the Settlements on the west side of the Mountains & put them to Flight; but I sinciarly think their Situation at Present very Dangerous their is upwards of 2000 Fameleys settled on that side of the Mountains, wich, I think, would be Cheifly cut of Shoud the Break out. On my arrival at Fort Pitt, I gave the Commanding Officer, & Mr. Croughan, all the Intelligence I had recived at the Indian Towns, who told me the had been Aprized of it before, & had Acquainted the Generall & Sir William of it. Mr. Croughan sent Mr. McKee down to the Lower Towns, in order to find out their Schemes & to Flusterate their Designs.

From Tobias Smollett, The Expedition of Humphry Clinker (1771):

A fine gentleman with a pig's tail, and a golden sord by his side, came to comfit me, and offered for to treat me with a pint of wind; but I would not stay; and so, in going through the dark passage, he began to show his cloven futt, and went for to be rude: my fellow servant Umphry Klinker bid him to be sivil, and he gave the young man a douse in the chops; but I'fackins, Mr. Klinker wa'n't long in his debt—with a good oak sapling he dusted his doublet, for all his golden cheese-toaster; and fipping me under his arm, carried me hoom, I nose not how, being I was in such a flustration. But, thank God! I'm now vaned from all such vanities; for what are all those rarities and vagaries to the glory that shall be revealed hereafter? O Molly! let not your poor heart be puffed up with vanity be puffed up with vanity.

From Dennis Lawler, Vicissitudes in Early Life; or, The History of Frank Neville, a Serio-comic, Sentimental, and Satirical Tale, volume 1 (1808):

"Dear me, Sir—to be sure it was a strange mistake; but really I feel so flustrated, that I hardly can speak—the young gentleman that is just gone out, Sir? I really don’t know who he is—he has lived with me near a month. When he first came, he was just come out of the country—he is very civil, and obliging, and generous as a prince; though I am afraid his circumstances—but, to be sure, I ought not to say any thing about that; for he always pays his way very handsomely: yet, poor young gentleman!—I am afraid he has something on his mind; for he seems quite molloncholy like."

And from "Court House, Old Bailey: The King, On the Prosecution of the West India Dock Company. versus John Smith, Samuel Hucks, Walter Foreman, and Daniel Hall (December 13, 1821) [combined snippets]:

Mr. Curwood. Now you are so important a witness—I believe you have been used to Courts of Justice?

A. I never was here before.

Q. Not at the Old Bailey?

A. Not at the Old Bailey.

Q. Perhaps you were never in any Court before?

A. I have been in the Court of Chancery.

Q. The Court of Chancery! I do not mean that Court, try again, have you been in any other Court before ?

A. Why, Sir, I am but a single individual, and if you flustrate me, possibly you might draw that from me by which I might perjure myself. I have been at Hicks Hall.

Q. I would not flustrate you for the world ?

A. You are extremely kind, and I am extremely obliged to you.

And from John Kennedy, Horse-shoe Robinson (1835):

"Tut, man! it ain't worth the trouble of talking about it; I never saw one of your people that I didn't know him by the first word that came out of his lips. You are an Englishman, and a red-coat into the bargain, as we call them in these parts. You have been a sodger. Now, never bounce at that, man; there's no great harm in belonging to that craft. They listed you, as likely as not, when you was flusticated with liquor—and you took your pay; there was a bargain, and it was your business to stand to it. ..."

Is 'flustrated' a portmanteau word today?

Historical support for the notion that flustrated is a true portmanteau word combining the central notions of flustered and frustrated is quite meager. But that's no to say that no one uses the word in that mashed-up sense today. Enough people have made unsubstantiated claims that flustrated is a hybrid of the two words that it would be surprising if no one used it that way.

One vignette-based assertion to this effect appears in Gary Schmidgall, editor, Intimate With Walt: Whitman's Conversations With Horace Traubel, 1888–1892 (2001):

Unsurprisingly, for one who was eager all his life to associate with what he called the "lower orders" ofsociety, Whitman delighted in slang. ...

The Mickle Street conversations show the poet a brilliant slangist himself, as well as an inveterate twister-up of familiar words and a neologist. A librarian can become a "bookophite," and "publisherial" manners can become the object of scorn. The usual raft of autograph requests, he will say, "infuriates me," but then with a laugh: "though the infuriation is not very violent." He could look about his cluttered bedroom and find "a devil-may-care-ness everywhere." Horace, in argumentative mode, becomes an "argyfier." He ridiculed the "flammeriness, the tinsel" of the South. Narrating a colorful Emerson anecdote, by combining "flustered" and "frustrated," he creates a word that deserves frequent use: "flustrated."

Even if Whitman narrated his Emerson anecdote in 1888, he was almost 200 years too late to be credited with creating flustrated. In fact, he was probably well aware of the word's prior existence, even if his editor writing in 2001 was not.

Timothy Findley, Journeyman: Travels of a Writer (2003) [quoted text not shown in snippet window] explicitly embraces a portmanteau-like meaning of the word:

flus-trate: n {fluster + frustrate): to produce a state of nervous tension through confrontation with something that does not function; usually denoted by the biting of lips and the making of fists. This birdhouse kit has left me so flustrated I ...

And likewise, Jack Butler, Dreamer (1998):

"It's just—I'm scared, so I'm making mistakes, getting flustrated." That was her neologism, her portmanteau word. Flustrated. She knew he liked it. "I'm scared I won't get in Williams." Now the self-deprecatory, ironic smile. He liked that too. She had learned that one from him. "I'm scared I will."

According to the website Everyone Is Jumping Off the Brooklyn Bridge, the 1995 edition of The Random House Webster's College Dictionary "states that flustrated originated in 1710-20 as a blend of 'fluster' and 'frustrate' (+ '-ed')"—but if that dictionary does so, it is breaking with the position established by its predecessor, The Random House College Dictionary (1975/1984):

flustrate v.t., -trated, -trating. Informal. to fluster. {FLUST(E)R + ATE, modeled on frustrate} —flustration, n.

Here, the dictionary explicitly equates flustrate with fluster; as in the entry for flustrate in Webster's Third New International Dictionary, the reference to frustrate appears only as a reference for the design of the word, not for its meaning.

Conclusions

The words flustrate. flustrated, and flustration (in various spellings) begin to appear in print by 1712. The initial meaning of flustrate seems to have been associated with drunkenness or tipsy befuddlement, but later senses of the word emphasize its equivalence to fluster.

More recently, some users have intended to combine the meanings of fluster and frustrate in their use of the word. Some users may even see flustrated as being primarily equivalent in meaning to frustrated, as in this example from Donald Leu, Charles Kinzer & Kathleen Hinchman, Literacies for the 21st Century: Research and Practice (1996) [combined snippets]:

Hindrance of Carl's learning. There were times when the help others offered was, perhaps not really helpful at all to Carl, but the family was still trying to be supportive. After only a few weeks, everyone was, as Carl's mate, Brenda, stated, "flustrated." Their emphasis on phonics seemed to be one source of frustration. They all insisted on having him sound out words most of the time, but Carl "tell them to hurry up and tell him the word," Brenda said. They seem to have agreed, however, that this was the best way to do things. Brenda explained, "Tony say that he never learn if they just tell him."

Even so, most current dictionaries that cover the word at all continue to treat flustrate as equivalent in meaning to fluster.

Going back to the earliest occurrences of flustrate, we may echo the OED's characterization of usage as "vulgar or jocular" (or as we might put it today, sincere or facetious). Of the nine historical instances from the period 1712–1835 discussed earlier in this answer, seven use the flustrate-family word as a distinct social marker, usually of lower-class sensibilities. Only two use the word un-self-consciously: the letter from the government agent writing to Governor Penn in 1771 (which is also by far the most "frustrate"-like of the early instances in intended meaning), and the testimony by a company spy in the West India Dock Company's prosecution of four suspected strike leaders.