Would it make sense to accept universal background checks in exchange for national reciprocity on state issued carry permits?

Sounds good. I'm not sure why that isn't the case already. Your driver's license and marriage certificate from one state are recognized by all other states; you don't have to reapply when you move. I don't see why gun permits would be any different. Article IV, Section 1: "Full Faith and Credit shall be given in each State to the public Acts, Records, and judicial Proceedings of every other State." Recognizing this in exchange for universal background checks would be a great deal.