States generally have sovereignty over criminal matters, so long as there is not a constitutional principle involved. States are not obligated to recognize other states’ laws if those laws do not conform to their own laws. (Just because State 1 gives a drivers license to a pig doesn’t mean that 49 states are obligated to let pigs drive in their state.)
Clearly part of this power involves setting policies toward what that state thinks are acceptable standards for legally carrying firearms in public. If a state thinks that another state issues permits in a way they do not think meets their standards, the state is currently free to say, “Nope, that’s not up to our standard.” The full faith and credit clause allows states to do this.
What you have been arguing is that states must accept the policies of other states with respect to these permits, even if the other states have very loose permit laws and the state in question has very strict permit laws. That’s not fair on several constitutional principles, even if you think everyone should carry guns with little regulation. It means the Federal government is substituting its judgment for some states for matters that are not within the Federal perview, and even more, subjecting the lawmaking power of a state to the lowest common denominator of any of the other 49 states.
To use a flawed analogy, let’s say conservatives get a bug in their bonnet about state taxes, and direct that states can only collect sales taxes only to the amount authorized by the home state of a nonresident (e.g., if an Oregonian buys something in California, the Oregonian pays zero sales tax). Such a law is wrong on multiple principles, but it’s pretty clear that the only point of such a law would be to subvert states that have perfectly constitutional tax laws that people in other states don’t wish to comply with.