I think the primary argument would come from Article 4 Section 1:
Full faith and credit shall be given in each state to the public acts, records, and judicial proceedings of every other state. And the Congress may by general laws prescribe the manner in which such acts, records, and proceedings shall be proved, and the effect thereof.
But then you have medical licenses and bar admissions...neither of which is recognized across state lins. And the state licenses marriages...they are not strictly a private contract between two (or more) people.
The founders were some pretty sharp people, but who could have foreseen—after thousands of years of tradition—that an attack against the sanctity of marriage would ever come from a portion of the populace rumored to be no more than 2%,and that 98% of the remainder would be forced to accept a mental aberration as normal.
This isn’t about whacko’s who are in love with each other this is about scamming for monetary gains that are denied to people who engage in perversion.
Why do we have laws forbidding more than one wife when we are accepting homoexual perversion. Certainly more than one wife is more normal, than what this is proposing.