There seems to be some inconsistencies when purchasing guns at a gun show or from individuals. One thing that could be done is a unified purchase procedure.
Tracking of violent or folks routinely suggesting violence, including mandatory reporting by mental health workers seems reasonable too.
Require anyone that has a war weopon, AK 47, AR 15, to have the same permits as someone that has a fully automatic weopon. Same for high capacity pistols. If they are caught outside their homes with such weopons and no permit, the weopons are confiscated and destroyed, and they face prison time.
You store your gun irresponsibly, someone takes it and commits a crime, you own that crime.
Not just reporting. Treatment. Without consideration of the ability to pay for it. The cost to society for untreated mental illness far exceeds the cost of treating such people.