By analogy, I could imagine a "Dormant Copyright Clause" doctrine, meaning that the states shouldn't have the power to legislate copyright other than in whatever contexts the Federal government explicitly leaves to them.

This is all theory, of course. But actual case law does say something at least similar. See for instance Sears, Roebuck & Co. v. Stiffel Co., a case in which the Supreme Court said (in the context of patents) that the Constitution reserves the power over them to the Federal Government exclusively, and that the states can't give patent protection to something that Federal law doesn't protect.