This makes sense and would not intrude on anyone's constitutional rights. The problem, however with universal background checks is a cultural phenomena which would require more of sellers, especially sellers in gun shows to purchase equipment not presently required of them. A solution would be to have entrance background checks on anyone who enters a gun show. This would decrease the responsibility on individual sellers and place more responsibility on organizers of gun shows who could charge an averaged fee to sellers to cover the cost of the new equipment. Anyone who is not legally allowed to purchase or own firearms should not be allowed in the gun show.
Of course, this could be left up to organizers to arrange how they see fit which would empower individual rights without government interference.
I have not seen individual legislation on congress.org which suggests such a proposal.