We could implement a truly universal background check. Let any person run a background check on any other person for any reason and get basic info on them such as any warrants, prohibitions, such as guns or to stay away from children, and so on. People could run them on blind dates, employees, babysitters, neighbors, and so on. They could be done by phone, online, email, fax, snail mail, all for free. You could get a receipt showing that you did the check and what it showed at that point in time to indemnify you for later. This system would take the wind out of the sails of the anti gun people's efforts and hopefully end the debate on background checks. At the same time, we need to implement far better protections for those who are put on the no gun list for reasons that aren't their fault, such as because the FBI messed up their SSN in their info and refuses to fix it without a court order.
The other problem with background check laws is that no one is or even wants to enforce them except occasionally as add-on charges.