What do I do if my supervisor insists that the classical CLT is false and wants me to write that in my paper?

(1) Would you say that my supervisor is scientifically incompetent?

If he says the CLT as stated and proved in textbooks is false, he is obviously wrong. "Scientifically incompetent" sounds like a very sweeping statement that I would avoid endorsing given the information currently available. He could simply be misunderstanding something in the language of the paper, or his knowledge of rigorous math could have an embarrassing gap (wouldn't be the first time), or it could be a failure of communication between you where you have a subtle misunderstanding of his position.

In any case, textbook CLT is correct, no question about it.

(2) How should I handle such a situation?

An honest intellectual dialogue with him to clarify the issue would be ideal, if he is the kind of person who can have such a dialogue when he is in the wrong and not back himself into an emotional corner and become upset and unable to reason. If you think that's risky, try to find another professor you trust and can consult about the issue, and if they agree with you, ask them to participate in a discussion with the advisor to make him understand your point. And try approaching any discussions with an open mind, in case it's you who are in error about what his position is exactly.

Finally, it's possible that your advisor really is scientifically incompetent, so if you are unable to make him come around to your view and conclude that he is intellectually unsuited to be your advisor, you should probably switch advisors. In any case, under no circumstance should you concede to his demand to deny the truth of a standard formulation of CLT in your paper. Good luck!

Be absolutely certain about what you both mean here. CLT has a very well established proof. It is almost inconceivable to think that a disproof of such a fundamental mathematical theorem would ever be published in PRL over a pure mathematics journal.

This leads us to infer, quite justifiably, I think, that the PRL paper must be addressing a popular application of CLT in some field of Physics. The conclusion here would then be that CLT is an inappropriate tool to analyze the system under study - not that CLT itself is invalid. It's not uncommon at all for Physics to change its mathematical tools from time to time as more is learned about the systems those tools are meant to model.

I would suggest that you take the time to sit down and discuss this with your supervisor until you both have a clear understanding of each others' ideas. Confrontation is not going to be productive here. If you can't clearly express your ideas to each other then the likelihood of producing a quality paper is likewise quite low. Sorting out this misunderstanding will likely do both of you some good.

I think that some supervisors may try to "teach" you something, which they perceive as passing on some sort of "knowledge" of theirs.

However, it may turn out (as it probably did in your case) that this "knowledge" was simply a hunch, a misunderstanding, or simply an uncompleted thought. Maybe they (half-heartedly) intended for this to start a discussion that would result in a new, revolutionary way of thinking about the CLT. I say half-heartedly, because they didn't even explain their idea.

I think there are many more and better ways to assess the scientific competence of your supervisor, like scientific output, quality/impact of lectures, etc. If he is indeed not suited to be advising students on how to do science, it should be apparent elsewhere (or even everywhere else).