DOMA or not its the opinion of most legal scholars that the Full Faith and Credit Clause will force states to honor each other’s marriage licenses. if two people are legally married in any state, then any other state must recognize the validity of that marriage and extend to them any benefits given married status in in that state.
Now because marriage confers a status, it is somewhat different than, say a driver’s license, which confers a privilege to do something within the boundaries of the licensing state (other states do honor out of state driver’s licenses but that is by interstate compact not FF&C).
A license to carry seems more like a driver’s license than a marriage one.
However, the real question you need to ask is WHY are licenses to carry Constitutional in the first place? If we follow the logic of Heller and accept that carrying is a personal right, like the right of free speech, then by what right does the state require a permit to exercise a fundamental right? Would the courts find it acceptable if the state wanted to issue a restrictive or burdensome “free speech license”?