Full faith clause has nothing to do with this since it applies to civil claims and court rulings by another state....Allen is proposing congressional legislation.
That's mostly how it's been applied, but there is a power of Congress in there, and it covers more than just civil claims and court rulings.
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.
Now it seems to me that a CHL law, the granting of licenses/permits, are public acts, with the records of who has a CHL being a public record (even if not available to the general public), although I'm sure that they were thinking of land deeds for the most part.
Now I agree that Congress could not mandate a CHL program be adopted by every state, but it seems there might be room for them to require each state to recognize the CHLs of other states, which is what I thought was was in Allen's bill.