Now, would state rights people be against national reciprocity?
Probably yes.
If nothing else, because it'd bring gay marriage reciprocity to the forefront. Some states legalized it, some don't. A4S1 doesn't necessarily cover CCW, because it's a permission slip for a certain activity. I'm not sure so sure CCW would be covered by "public acts, records, and judicial proceedings" unless a state had the audacity of handing out CCW permits allegedly valid for the entire United States (ie redo the structure of CCW licensing to be like marriage licensing). Every state CCW/LTCF law I'm aware of tends to be very strict in saying it is only valid for its own state, unless another state decides independently to accept it.
IMHO, it'd be nice if there was a standard CCW framework. Maybe set it up in Tiers that states can agree to. Tier 1 requires nothing, Tier 2 requires a background check and reasonable costs, Tier 3 requires training, high cost, fingerprints, etc. This would be independent of any license the State wanted to hand out. Never happen, but it's logical. We almost have it with Utah and/or Florida permits. Having widely varying CCW laws kinda does suck.