I hate hypocrites, homophobes, and religious nuts. So yes, if a gay-hating religious ass is gay, I would be fine with, and fully support outing him.
Talks of privacy or respect is cheap. When someone is actively trying to hide behind a wall of hate while still trying to secretly enjoy the object of that hatred, and making it difficult for others to enjoy it, then that’s an attack and deserves to be countered.
Nobody would say that if a person is lying about something, like taxes, that it would be wrong to out him as a tax cheat. All of the misplaced sympathy is for gay people as a whole, who have it more difficult than straights, and so people who are gay and like gays have this misguided notion of camaraderie with the gay homophobe. Some of them think that by killing them with kindness, they can get them to change their mind. However, being gay doesn’t automatically make you a good person. These hypocrites are as odious as they would be if they were straight. So out them, hit them where it hurts, turn their hatred as a weapon on them. Either they will be hurt by it or change their mind, but either way the result is positive