I have to wonder if the mental health “evaluation” (and how THAT is going to be run is a whole can of worms unto itself, plus whether it is even Constitutional) runs afoul of federal law under HIPPA. If it does, there’s a valid challenge right there.
I don’t see how it would run afoul of the privacy act. They’ll simply require you to sign a waiver as part of the application process. Remember, in New York the second amendment is a privilege not a right.