Full faith and credit doesn't apply here. A license issued by a state is an exception to the laws within that state.
However, Congress could do a great deal to restore our Second Amendment under Article I, Section 8, Clause 16 (providing for the militia). They could simply find that having an armed population is beneficial for the security of the country and its citizens, and on that basis pass laws nullifying any state law that exceeds certain boundaries. Congress could, by the authority granted by this clause, supercede all state laws infringing on the Second Amendment.
Maybe the simplest way is for Congress to pass a law requiring any state (or political subdivision thereof - cities, etc.) to recognize any license issued by another state provided that the application process for the license meets certain requirements (criminal background check, proof of some minimal level of training). Yeah, I know, residents of states like PA that don't require training wouldn't benefit from their home state licenses and would have to get nonresident licenses in other states, but that is really the most likely way to get the statute passed. Worded this way, a NJ resident wanting to carry in NJ could do so by getting a FL or VA license.
For those of us who live in states with relatively friendly laws, it might be a mixed blessing to have these issues decided on the federal rather than the state level, especially if we end up with the Demagogic Party controlling both houses of Congress and the White House. However, given the current trend towards more relaxed gun laws just about everywhere other than the People's Republic of Kalifornia, such a statutes just might be the best way to bring the sure holdouts like DC, CA, NJ, MA, MD, IL, NY, etc. into line with the rest of the country.
