Those adjudicated mentally ill are already prohibited from possession of firearms.  If you think that is not enough, how would you define who is mentally ill? Who would set the standard? How would we test people? Would agree to using the same test before allowing a mentally ill person to use twitter? Or the DP comment section? If someone is determined mentally ill, can they get off that "list"?  You will be hard pressed to get anything like this in place as it violates peoples civil rights.

As for background checks on private sales - that is universal background checks and it's a horrible policy. It's only enforceable with universal registration, which won't happen. It goes too far and violates other Rights like 4A.