No power is enumerated allowing for registration.
Of course, your opponent will cite numerous court cases that allow the fed to do pretty much anything it pleases under the interstate commerce or general welfare clauses.
At that point the question becomes: Are government powers solely granted by the People through the Constitution, or do the various branches of government grant themselves powers independent of any constitutional restraints, limited only by the ballot box?
The answer to that question decides what type of country we live in: a constitutional republic where individuals have unalienable rights, or an unlimited democracy where individuals have only privileges the fickle majority grants them.
Just FYI, Congress only has the power to tax and spend for the general welfare, not legislate for the general welfare. Things like "general welfare" are delegated to the states under their so-called "police power" to regulate for health, safety, morals, etc.
I agree, though, that the feds have been given free reign to do darn near anything under the guise of "commerce," and we have an intellectually corrupt federal judiciary who is happy to give the feds whatever powers they seek under the Commerce Clause.