You see an answer
The answer is generally good. Extensive research, in-detail review of the claim topic, references, etc...
However, one small aspect of the answer is awful - doesn't have solid references (but has enough to avoid being edited out by moderators), and/or uses logical fallacy, and/or can be refuted by better references
Basically, if that small aspect was the whole answer, it'd be a clear "downvote!" material.
moreover, despite the scope of the bad content being small, it is important - either in isolation (e.g. it's denying Holocaust, or defending slavery), or even worse, because it is an important data point for the overall answers' conclusion, and refuting that data point either fully invalidates the conclusion, or makes it less reliable.
The poster, upon requesting that the "bad" nugget in the answer be addressed, refuses to do so, explicitly - and they (or moderators) refuse edit suggestions - rightfully - because the edit contradicts the poster's intent.
What should be done in such a situation to address it effectively?
It seems that 100% of actions I'd consider as possible are either wrong or not effective (based on my extensive practical experience on this site and SE overall):
Upvoting the answer has a clear downside that it rewards the "bad nugget" content and countra-incentivizes the poster from fixing it.
Downvoting the answer is not terribly fair (the good content largely outweighs the bad) - and more importantly, is not effective since a large majority of the users upvote said good content and ignore the bad content
Informing the poster, as noted in the preconditions, was not effective. Ditto for trying to edit.
Flagging for moderators is a wrong action since the bad content is not moderator-action-awful grade.
posting your own counter-answer refuting bad content seems bad/useless - the scope of the "bad nugget" is small enough that merely restricting your answer to that nugget would make your answer "not an answer" to the main claim, or at best, a mediocre answer due to small scope. And anything above that small scope is already 100% answered by the poster in their answer, well.
posting your own counter-answer which is a clone of full original answer with bad nugget fixed seems at best awful and at worst plagiarism.