It can be determined if one has pneumonia (antibiotics for the pneumococcal form) or measles, or epilespsy seizures requiring anti-seizure medication, or a broken arm, or cancer type and stage. But the state and quality of one's mind and emotions iss not so easily measured or predictable. Unless one has been comitted to a mental hospital, or has a criminal record of unlawful behavior (indicating some degree of departing from a tolerable range of actions), leave the background checks alone.
However, since most of the shootings have been a result of alcohol or psychoactive drug intake, make the personal sales to, ownership of, or or personal carrying of dangerous weapons a felony if one's BCA is greater than 0.1% or if ones mentality is dependent on psychoactive medication. That's measurable, quantitatively.
May be worth a thought, I dunno if it would fly. But "psychiatric" evaluation score? No. Don't go there.
Not so sure about that. I'd say "suicidal" gives you a definite F.