In the sentence “you have a problem”, “have” is the main verb. When reduced to the clitic “'ve”, it becomes a weak form and is only expected to be used as an auxiliary verb. These types of verbs must be followed by the main verb. “a” is not a verb. Thus, we insert “got”.

If we do not insert “got”, the stress in the sentence moves and it sounds overly affected.

Why this is the case here and not for other verbs, like in “You’re a man”, I’m not too sure though.