A single background check, much line the one done when you get a handgun permit in most states, would not be an issue for me.
The best idea I've heard in a long time came from a retired Master Trooper from the Indiana state police, and a long time MP from the National Guard/Active Duty (Three tours in the sand box)...
Right on your drivers licence,
in Green if you are legal to own handguns, high cap mags, ect.
FC (Firearms Cleared)
CC (Concealed Carry)
HT (Hunting & Target)
FR (Firearms Restricted To Long Guns)
Then in Red,
FP (Firearms Prohibited)
FF (Firearms Felony)
Give the power to the Judicial System to confiscate the drives licence on the spot when a potential felony or mental illness is addressed, just like what is done right now in many states with DUI/OWI (Driving Under Influence/Operating While Intoxicated) and many states confiscate handgun permits in the cases of domestic battery or other potential issues.
With a glance, this would give gun shops, private sellers, ect. and INSTANT way to be reasonably sure they are NOT selling to prohibited persons.
This would also give law enforcement an INSTANT way to tell if that hollow point bullet they just found in a felon's vehicle is a misdemeanor or felony, and if they should do further investigation/take more precautions.
This wouldn't interfere with the 'Rights' of a mental patient by singling them out as such, it would simply give the people that should know a way of identifying the folks that SHOULD NOT have firearms.
In our state, it's a one time fee of about $160 to have the background check done and get a lifetime Handgun Permit, and I'd be glad to do the same if it would stop 'Gray Market' sales to people that should NOT have firearms in the first place.
The last thing I want to do is supply the firearms, ammunition, ect. to some guy that uses it to kill people...