"he wants us to stop shouting at each other"

What this refers to is the exact opposite of how you are using it against Bernie. He wants us to stop using gun policy as a wedge issue (which is what you yourself are doing) so we can actually enact reforms to work on the problems you address, rather than just doubling down on the two parties' entrenched positions, which each side uses for political posturing, and that gets nothing done. That's what Bernie's statement means.

I hate guns personally, but I don't think they're the primary problem, they're the weapon of choice for desperate disempowered people.

I support any sensible gun regulation. Suing gun manufacturers for the wrongful use of their products doesn't seem too reasonable to me (I know your OP didn't mention this, but it is often used against Bernie that he doesn't support this).

I'd support a waiting period, ending the gunshow loophole, a gun registration database, etc.

I'm a little concerned about mental health criteria. Most gun violence happens when violently inclined people act out in rage or in crime. I don't think there's a high correlation to any reliably diagnosed psychological disorders.

The real problem that needs addressing is that people in this society, many of them, are in huge trouble in their lives, and there is no positive societal mechanism to proactively work with such people. Most gun violence could be prevented by such programs. For example, why not have funded support programs for anyone who has income insecurity, whether they qualify for unemployment or disability or not (most people fall through the cracks and are receiving neither benefit).

Or a funded support program for people going through life crisis, whether they lost a child, are having difficulty in a relationship or dealing with ending a relationship, there should be many categories that would be eligible for support. This support should include a mechanism to actually plug people into positive social environments (sports or crafts activities, or work apprenticeship opportunities, why not have social organizations for the recently single, or for people dealing with dependencies). Everything is crisis and punitive focused, rather than a caring system that identifies at-risk people before a crisis and actually tries to address their needs.

Try that, and watch gun violence go down, I guarantee you it would work. These are the kind of things a country like Norway is probably better at doing than our own country.