Any psychological background check will have hazards, no doubt. And I agree that some probably would avoid help given that situation.
As far as the check itself goes, however, it would require reforming HIPAA law to allow for the check to access personal information. The diagnosis of mental illness would not be connected to the background check itself, because, as you said, there is the hazard of political bias. Diagnosis of something like schizophrenia would still be something that would be determined by a person's psychologist -- someone they trust, not a potential political activist type.
All that aside, I disagree with a lot of proposals by Democrats on this issue.
As the late Rush Limbaugh used to say, I know liberals like I know every square inch of my glorious naked body. That being said, I think back to when the left was pushing for gays to be out of the closet. That's all they wanted, and they'd be happy. Today, they forced their marriage down our throats, and even are able to adopt children.
I remember at a time where everybody smoked. The left asked that it be restricted in movie theaters. That's all they wanted, and they'd be happy. Today, you can't smoke a cigarette in most public places, and in blue states, not even outside in a park or on the beech.
What I'm saying here is that Democrats never stop at point A. Oh, they say that's all they want at the time, but after they reach point A, they proceed to point B, then C, then D...........
Once we agree to psychological background checks or exams, we'll be opening up a new Pandora's Box.