I have a friend who’s a social worker for Catholic Social Services. They know she is not Catholic and never will be.
That said – part of her job is assisting low income women/families… she is not allowed to bring up the topic of family planning at all. She does answer questions when asked specifically – but cannot volunteer information of any kind. That’s part of working for a social services agencies that is managed by the Catholic Church, but funded in part, by the US government. Obviously birth control is legal, and there are clinics that make it available at low cost for low income folks – but she can’t talk about it unless very directly asked. Even then, it’s discouraged. If the agency were privately funded, then they could even eliminate answering a direct question.
Now – to the question at hand. I don’t think any employer should be allowed to discriminate based on race, gender, religion, or sexual orientation. Obviously there are exceptions to this for actual churches (not just church affiliated agencies) – but churches themselves, don’t get federal funds. The Catholic church is free to ban women from the priesthood, etc. They are free to disallow gay priests (yeah, good luck with that one.)
But employers who are not actual churches should not be allowed to discriminate – and that should include gender identity. They can establish dress codes etc to make sure that gender identity is not overt… as long as it is applied evenly to gay and straight alike.