A basic background check doesn't cost anywhere near "thousands" per person. We aren't getting people top secret government clearances.
You are doing a background check that requires three different agencies. You are required to do fingerprints and a photo. Those things cost time and money. I did some searching and could not find a estimate for how much each NFA background check costs, but I did find some small piece of information that raises my estimate of how much they cost: It is not a one time deal. You have to then support each NFA stamp by requiring approval for every transfer, ever time someone crosses a state line, every time they change address. So, unless you want to charge for all of that as well, it needs to all be factored into that original fee.
It seems that putting them in the NFA (which would also tie finger prints to serial numbers) would be a good first step.
I agree it is a good first step. It has a number of other provisions I think would help as well.
It makes the penalty for owning a NFA controlled item considerable higher than is current for any other firearm and with less burden of proof. Just having possession of a NFA controlled firearm with out proof of a permit is a felony with a $10,000 fine and 10 years in federal prison.
You can not sell or otherwise transfer a NFA controlled item with out ATF permission and a background check, basically all transfers of NFA controlled items is treated just like an original sell. It closes the gun show loophole and makes all grey markets into a black market.
You can not cross state lines with a NFA controlled item without written permission from the ATF.
You must always be able to provide proof of registration for any ATF controlled item on request. (want to carry a gun, you must carry it's registration papers as well)
Obliterating, removing, changing, or altering the serial number of a NFA controlled item is against the law.
Any violation of those things could carry a penalty of up to $10,000 and 10 years in federal prison, and the possibility of secondary charges that include of up to $500,000 fine and an additional 5 years in prison.