How does existence preceding essence not preclude the possibility of bad faith? In what sense is it possible to act inauthentically if there is no authenticity other than what we make for ourselves?
I mean, it's absolutely possible to deny that existence precedes essence. A lot of people aren't existentialists. It's pretty central to Christian doctrine, for example, that essence precedes existence. The issue I have with rejecting it is that it's the original and compelling claim that existentialism is making.
I guess you could make some kantian-type transcendental argument about the limits of freedom, you can't freely choose to be not free, etc. etc., but bad faith doesn't seem as narrow as that. It vaguely feels like 'making choices Sartre thinks are beneath human dignity', but perhaps that's uncharitable of me.
Philosophers accuse each other of being dogmatists all the time, or what have you, so it seems ok in my book that Sartre would call an essentialist Christian inauthentic if he is making a universal claim against a priori essence.
Though, I'm not sure Sartre would rely on dignity exactly, but more like these are just the facts of the matter: there is no a priori essence with which you can operate. Denying the facts would be tantamount to lying, lying sort of like bad faith. Living inauthentically might be like not being aware of the facts.