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I'm writing a letter criticism to a local newspaper for publishing a letter expressing climate change denial.

I have this line I'm struggling to find a word for:

In the era of fake news and post-truth, the newspaper would do well to exercise some journalistic [something] and make it clear what the truth about the matter is.

Here, what I'm getting at is that the newspaper has both the expertise (in being truth gatherers) and ability to do a thing, though they don't have the obligation to do so.

In other contexts you might say:

The cook should exercise his culinary [something] and insist he be provided with the finest olive oil.

The doctor should exercise his medical [something] and tell his patient that the rash should be treated, even though she came in for a runny nose.

the [adjective describing the area of expertise] [something] format isn't necessary for the word I'm looking for.

You could choose integrity, defined in Mirriam-Webster as:

'firm adherence to a code of especially moral or artistic values'

Or probity, defined (again in Mirriam-Webster) as:

adherence to the highest principles and ideals

Either of those would imply that the newspaper should follow the ideals and standards of the industry. In the UK that would be the Editors' Code of Practice. The relevant part of the Code (section 1, on Accuracy) says:

1. Accuracy

i) The Press must take care not to publish inaccurate, misleading or
distorted information or images, including headlines not supported by
the text.

ii) A significant inaccuracy, misleading statement or distortion must
be corrected, promptly and with due prominence, and — where
appropriate — an apology published. In cases involving IPSO, due
prominence should be as required by the regulator.

iii) A fair opportunity to reply to significant inaccuracies should be
given, when reasonably called for.

iv) The Press, while free to editorialise and campaign, must
distinguish clearly between comment, conjecture and fact.

v) A publication must report fairly and accurately the outcome of an
action for defamation to which it has been a party, unless an agreed
settlement states otherwise, or an agreed statement is published.

Your sample sentence would then be:

In the era of fake news and post-truth, the newspaper would do well to
exercise some journalistic probity and make it clear what the truth
about the matter is.

The word professionalism is defined in the Oxford online dictionary as

The competence or skill expected of a professional.

This definition, as it includes the word expected, implies the commitment and dedication to actually do the job properly. It would be quite possible for a skilled decorator, for example, to paint a room poorly but that would be unprofessional; particularly if he was being paid to do it properly. Similarly journalists who do not make the effort to check their facts can be accused, with justification, of acting unprofessionally. "Professionalism" is quite suitable in your sentence.

credentials

The cook should exercise his culinary **credentials** and insist he be provided with the finest olive oil.



The doctor should exercise his medical **credentials** and tell his patient that the rash should be treated, even though she came in for a runny nose.



With fake-news rampant, the newspaper would do well to use its journalistic **credentials** and make clear the truth about the matter.

Definition of credential in English: Ox Eng Dict

noun usually credentials

1A qualification, achievement, quality, or aspect of a person's background, especially when used to indicate their suitability for

something.
‘recruitment is based mainly on academic credentials’

1.1 A document proving a person's identity or qualifications.

‘examine carefully the credentials of all callers before admitting them’

1.2 A letter of introduction given by a government to an ambassador before a new posting. ‘the Russian ambassador presented his
credentials on September 30’

Acumen noun : The ability to make good judgements and take quick decisions. ‘she hides a shrewd business acumen

To me, it then shows the expertise and ability to do something rather quick (which, again, shows the expertise).

Plus you can use it with any other word like 'journalistic acumen', 'medical acumen'...etc

Moxie noun

Merriam-Webster provides one definition of moxie as being,

3: know-how

You can use know-how as well. It means,

knowledge of how to do something smoothly and efficiently; expertise

I think moxie or know-how would fit in your provided sentences.