I'll helpfully refer you to the Full Faith and Credit Clause of the US Constitution. If Massachutsets has to honor the drivers licenses of other states and Massachutsets expects Texas to honor their phony "gay" marriage licenses then MA has to honor Florida CCW permits.
They may not have to honoor the MA gay marriages. Lots of the state laws explicitly say they will not recognize any gay marriages or anything akin to gay marriages (gay marriage by another name), period. And they haven’t been overturned. WI has such a clause in our law.