I am struck by the glib contradiction that underlies advocacy for background checks for firearms purchases.
Part of the contradiction stems from the oft-quoted recitation of "the right of the people to keep and bear arms shall not be infringed." "the people" doesn't have any modifiers, and is read elsewhere to apply to all people. Yet background checks are intentionally designed to infringe on the rights of many people, and there is no statutory or judicially-defined limit, as far as I know, on the number of people whose rights may be infringed by background checks.
The other part of the contradiction is that background checks don't work to limit access to guns by people whose possession of a gun should give you the willies. The opponents of gun control measures know it and prefer something that doesn't work to something that does.
In light of the beginning clause of the Second Amendment, I suggest that a better background check would be to verify that the purchaser of a firearm is a registered and active member of the State's full-time, regular militia, and that the firearm being purchased is on the State militia's list of ordnance-approved weapons. While some States also authorize an irregular militia, the members of the irregular militia should also be restricted to ordnance-approved weapons.
There is a substantial advantage to militia operations in having identical weapons carried by all members. Establishing an ordnance-approved list of weapons makes the procurement, storage, and delivery of compatible ammunition to all militiamen/women much easier. It improves weapon maintenance because parts are readily interchangeable. It accelerates firearms training since all militiamen/women are firing the same weapons. It improves the efficiency of procurement of additional magazines, scopes, cases and other accessories.
This is even more important when considering that the State militia may be called out for a national emergency and have to intermingle and operate jointly with militias from other States and with federal troops. Incompatible weapons can lead to lost battles and substantial collateral damage.
It is also important when considering that the militia may be called out for a State emergency and have to intermingle and operate jointly with State, county and local law enforcement personnel. Incompatible weapons can impede collaboration and hamper the response to an emergency.
While I haven't checked, my guess is that the military Reserve and National Guard units in all the States and territories use Department of Defense ordnance-approved firearms. These units are the current embodiment of the Constitutional militias mentioned in the 2nd Amendment.
As a practical matter, proving active membership in the State militia would entail nothing more than displaying the militia-issued identification card. Under this protocol, the real background check would be performed when a person enlisted in the militia, and at regular intervals during their term of service. The background check would satisfy at least two objectives - determining trustworthiness and character to qualify for a security clearance for appropriate access to classified information, and determining psychological characteristics that would foster or impede integration into and effective participation in militia operations.
This protocol would have the additional advantage of ensuring that all persons with firearms had received stringent training, carried, stored, displayed and used their firearm pursuant to the guidelines and rules of engagement of the State militia, and were subject to militia discipline regarding the use and misuse of their firearm.