You're exactly right, and a lot of people get caught up in the idea that they like a law, not whether the Congress has the authority. I love the idea of this law, but the Congress has no authority.
From the Constitution:
Full Faith and Credit shall be given in each State to the public Acts, Records, and judicial Proceedings of every other State. And the Congress may by general Laws prescribe the Manner in which such Acts, Records, and Proceedings shall be proved, and the Effect thereof.
Certainly the granting of a permit to carry a concealed firearm is a "public Act". Assuming that a state is justified in demanding that a citizen qualify for a permit, what is reasonable about expecting a traveler to qualify for a permit in every state visited? This clause of the Constitution was to prevent the very prohibitions we are seeing.
Those who believe that states should be able to disarm their own citizens must also believe that our Founders intended that states could disarm visitors from other states.
For much of our history the punishment for stealing a man's horse was hanging. This is because depriving a man of his horse could strand him and result in his death.
The punishment for disarming a law-abiding citizen should be the same, whether that citizen is a resident of the state or a visitor.
Tell me in more detail now why you believe the bill in question to be unConstitutional?