As other commentators have pointed out, it is important to establish exactly what your supervisor is asserting. It would be somewhat unusual to claim that a well-known and well-vetted mathematical theorem is wrong, without some specific argument constituting a counter-proof. Before you make any big decisions, make absolutely sure that your supervisor is objecting to the actual theorem (when correctly stated) and does not have an objection merely to a misstatement or misapplication of the theorem.
Now, supposing that he really is objecting to the CLT, as correctly stated, I think the next step is this. If he is of the view that a well-known mathematical theorem is false, then the onus is on him to clearly and rigorously show his reasoning and establish the correctness of his own view with the mathematical community. If he is right (he is not) and he can prove it (he can't) then his discovery would be one of the most impressive discoveries in the history of probability theory, and it would win him permanent acclaim in the field. It is up to him to publish his argument if he has one. In the meantime, there is a wide literature asserting proofs of the theorem that have been accepted by the mathematical community, with thousands of trained eyes scrutinising it for centuries. He has no legitimate standing to block your work on the basis that he does not accept the theorem.
I would suggest that you firstly bring the matter up with him in a polite but forceful manner, and insist that you regard the classical CLT to be correct, and you will not say anything to the contrary in your work. If he is going to use his vetting prerogative to seek to pressure you to say something that you (along with the rest of the mathematical community) regard as untrue, then you should bypass him completely and publish your work without his input. If necessary you could ask another academic to take over the vetting role, or simply bypass it altogether.
Lastly, I disagree a bit with some other commentators regarding the issue of competence. Even if your supervisor is wrong on this issue, I wouldn't assume he is incompetent in any broader sense (at least not on this evidence alone). Mathematical theorems can be hard, and the proof of the CLT is not trivial. It is entirely possible that your supervisor is a competent physicist in other respects and just has a "blind spot" in this aspect of probability theory (which is not even within his direct field). If it is remains feasible to work effectively with this supervisor despite your disagreement on this issue, and if that is the only thing you notice where he is in error, there is probably scope to learn plenty of other things from him.