I don't see how this doesn't force reciprocity between all states. Drivers licences, marriage certificates, and all other "acts, records, and proceedings" are recognized by all the states. Why not a carry licence?
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.