I'd actually support the "gun show loophole." Make all vendors who sell a gun hold an FFL. Sure, private sales can still occur elsewhere, but you've reduced the visibility and presumably lower prices that come with the environment.
I honestly think we should focus on exiting laws and upgrading technology, but I'm keenly aware this topic isn't going away. The more the Right and the NRA stand there with their arms crossed the more ground they lose. They really need to throw the Left a bone here and then dig in. Let the Left scream "it isn't enough" rather than "they are doing nothing." The Right can help shape the almost inevitable changes or get trampled, probably with stiffer consequences.