I agree gun permits should be free if not extremely cheap, but as SCOTUS has ruled that permits to possess a gun are Constitutional, I would imagine they would also rule a fee to pay for such permit is within reason as long as the fee directly corrolates to the cost of the permit. Which is probably no more than $20. Fingerprints should be free.Not just a registry, but an illegal tax too.
You can't charge a fee for exercising a right.
Poll taxes were ruled unconstitutional long ago.
So, even if there *were* some background check scheme that was permitted by the constitution (which there isn't), it would have to be at no charge.