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An example to clarify:

Say John made a drunken fool of himself at the office party, which not everyone attended. On Monday, at the company lunch, Mike says publicly, with feigned sincerity, "John, I just want to let you know, we've all been there, and there's no reason to be embarrassed today." With this, even the absentees will inquire what happened, and the story will be retold.

There ingredients here are:

1. Mike's intention was to highlight the episode with the aim of further humiliating John


2. Mike's literal statement is the opposite of his intention, since intentionally embarrassing John would make him look bad


3. Merely saying the thing is enough to accomplish goal 1. That is, what's important is directing people's attention to the subject.

Is there a word to describe this specific type of underhanded statement? Either a verb as in "Mike ____ John by pretending to comfort him at lunch". Or a noun as in "Mike's comforting John at lunch was an expertly executed ____"

Some clarifying notes

I'm after something far more specific than irony or misdirection. I'm really looking to name a specific social tactic for "innocently" directing attention to damaging information

Although I'm not entirely sure it fits your example, this sounds similar to me to the concept of "paralipsis": drawing attention to something by saying that you will not mention it. The Wikipedia article on "Apophasis" mentions a few synonyms, and also gives various examples which show that depending on the context, the actual intention of the speaker may be fairly transparent (that is, someone employing paralipsis is not always trying to be perceived as sincere; the goal might just be to establish some minimum level of "plausible deniability").

Specifically, I think that saying "there's no reason to be embarrassed" could be interpreted as a "paraliptic" way of communicating that there is reason for John to be embarrassed.

"Paralipsis" has been mentioned in the answers to various other questions on this site, such as "I'm happy to see that you are sober as a judge" Is this a rhetorical device?

It's close to a 'back-handed' compliment -

an insult that is disguised as, or accompanied by, a compliment, especially in situations where the belittling or condescension is intentional.

Without a source to quote, I'd suggest the phrase poisonously sweet, except that I've only seen the phrase applied to women.

It is not, exactly, ironic, since Mike's statement is not intended directly to convey the opposite of its surface meaning, but rather to provoke a social response which is intended to embarrass John despite the professed intent.

I can think of schadenfreude, which, according to Merriam-Webster, is a noun that means "enjoyment obtained from the troubles of others".

In general it is irony ("The use of words to express something different from and often opposite to their literal meaning." American Heritage).

If said with intention to hurt, it becomes sarcasm ("A cutting, often ironic remark intended to express contempt or ridicule" Ibid.).

See also What are the similarities and differences between "irony" and "sarcasm"?