OK, here’s a solution. Only go to doctors belonging to a “church” (created by the medical profession) to which the members (you) confess your “sins”. I really don’t think the government would cross that line, and if so, only once.
Christian therapists are being told they can’t refuse to treat homosexuals who don’t want to be homosexual (conversion therapy). Chris Christie was even proud to sign that law in NJ.
Nor in some states can they tell mentally ill who think they are an opposite gender to rectify mind with reality, in some states.
In these cases, the only legally allowed therapy is yeah, you’re homosexual, OK, you think you’re a woman, let’s start hormones and plan on surgery.
So no, the religious exemption to intrusive questions isn’t allowed.