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.
I’m not too sure, but I think “be” (“is”, “are”) is the only verb that can be contracted and still remain a main verb. I’m not too sure why.