In that case they should not label the disagreement as "descriptivist", because descriptivism is entirely incidental to the dispute (even if the person happens to be a descriptivist).

For a formal argument about descriptivism I'd agree with your "should". I disagree for a throwaway joke playing (as far as I can see) on an implicit understanding that descriptivists often go out of their way to rebut undue prescriptivism. (But I guess this is a side debate about our personal thresholds for jokers making a punchline land by relying on a word's connotation instead of its formal meaning.)

It is logically impossible to refute a prescriptive claim with a descriptive one; and one's interest in description cannot itself provide any support for one's prescriptive claims (which include disputes of others' prescriptive claims). If I say that "ain't" is an improper word, and you say "I disagree, because I'm a descriptivist", then you are misunderstanding what descriptivism means.

I might be misunderstanding something, because I think you're only correct given particular, narrow meanings of e.g. "improper" & "mistake". People often use words like these in another way: to make prescriptive claims that simultaneously put forward and rely on (whether explicitly or not) descriptive claims that can potentially be refuted by another descriptive claim. If I say "'ain't' isn't a proper word", I could mean a number of things. I might mean that "ain't" shouldn't be used because it connotes low status tout court. If so, pointing out a dialect or subculture in which it indicates high status would refute me. I might mean that "ain't" shouldn't be used because it's a neologism. Pointing out that it's an old usage would then refute me. I might mean that "ain't" shouldn't be used because it's difficult to understand. Survey data showing that most language speakers readily understand it would then refute me. These would be examples of refuting a prescriptive claim with a descriptive one.

Sure, strictly these aren't direct refutations of the prescriptive claim. But in practice some prescriptive claims live or die on the basis of some falsifiable descriptive claim. I suspect most prescriptive claims made by everyday people do; prescriptions that are just bald assertions are harder to defend.