> I'm no constitutional expert, but I'm having a hard time figuring out how the federal government could possibly have the authority this sort of law would require.
Requirements to make and keep records which demonstrate compliance with legal restrictions are a routine and essential part of the enforceability of many laws, so, to the extent that the Constitution explicitly imposes restrictions on the States and explicitly authorizes Congress to enforce those restrictions through "appropriate legislation" (especially in light of the necessary and proper clause), Congressionally-imposed recordkeeping requirements to demonstrate compliance would seem to be within the scope of the Congressional power thus granted.
This is relevant because the 14th Amendment commands that "No State shall [...] deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws" and provides that "The Congress shall have power to enforce, by appropriate legislation, the provisions of this article".
Even with the way the Supreme Court has limited the authorization clause of the 14th Amendment to (unlike identical language elsewhere) only authorize corrective legislation, so that it might not be possible for Congress to impose a general and unconditional requirement, Congress could still impose a requirement on particular states as a consequences of past violation of Constitutional due process or equal protection occurring under the power of those states.
(It could also do it by making it a prerequisite for, say, federal law enforcement assistance funding, or access to federal law enforcement databases, etc. -- and it could do both in parallel, so it would be required both for states found to have past violations for which it serves as a remedy, and for states wishing to federal assistance for law enforcement.)