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!