One way to approach that lie and bring it forward into the larger public sphere would be to say, "ok, you say mental health is the problem. Well, then you should support an expansion of access to mental health treatment options. Here's the bill we want you to support."
Or take a mental health funding bill that they dumped or reduced funds for, and say "if you truly believe mental health issues are the cause of these travesties, why have you refused to support mental health treatment for ALL Americans?"
PUSH their arguments to their logical conclusions. Like "Guns don't kill people. People kill people." Turn it around to "People WITH guns kill people."
They play games with logic. We can catch them at those games instead of arguing with their arguments. You will never meet a member of this Klan nor its supporters that has the slightest clue about logic.