I purged a long discussion not relevant for the question. Please read the post notice on top of the question and this FAQ before posting another comment. Recall also to keep a respectful tone.

Consider, at least, that other students may not come from the same culture and may not understand naming conventions and may need some explicit help in this.

Great question, but the legal part of it, you’d get much higher quality advice if you ask it on law.se.

Is it normal that a TA writes the course outline and not the prof?

Are you at a public or private school?

Related: academia.stackexchange.com/q/123994/40589

@Buffy Been said, been erased. ;)

Of course refusing to do something your professor is asking will (or at least could) affect your academic career. To which extent it will affect your career is something that will heavily depend on how important this is to them, how easily they can move past it, how long you work with them and how useful a good recommendation from them would be to you.

While we might think that professionalism trumps politics, the courtesies we extend to each others of a different perspective are not without limits. I would suggest that a prudent way to approach this (for the sake of your career) would be to show her that you're making an effort to accommodate her request while still respecting your own moral/ethical position. That doesn't mean giving in, but it does mean proposing alternatives and non-heated discussion. Human beings don't need everyone to agree with them, but we do need to feel like we're being taken seriously, at work and everywhere.

Also, to be frank, your question of "Am I legally obligated to do so?" is a bit of a red flag. Legal boundaries in society tend to be way past the boundaries at which your career is adversely affected. If you're at the point where the law is a limiting factor in your interactions with your supervisor I'd suggest re-evaluating your relationship and/or seeking work elsewhere. Good luck. I'm very interested to hear how this pans out for you.

It's disappointing that the mods are deleting on-topic comments - I put a lot of effort into that...

What would writing in the pronoun look like? Ms. Julie Bobset (she)? I'm curious about the actual demand here

What are your social and religious reasons? Are they important enough to come before the emotional comfort of your students?

@quant - 2 things. First: please see the post notice, comments that do not request clarification or suggest improvement will be deleted without warning. Second: as a courtesy, we normally move comments to chat rather than deleting them when possible (the system only lets us do this once). I thought we had already done this, but now I see I was mistaken. So, thanks for the heads-up: I have undeleted your comments (and others) and moved them here. Cheers!

@cag51 Thanks for the clarification.

@OldBunny2800 Students can call me by my name, please see my last comments. What about my ethical/religious/moral values? Why do you put me outside of the equation?

@poman putting myself in your shoes, I'd be replacing "Don't get me involved with this" with something like "I respect yours and everyone's views on this, but I'm deeply uncomfortable publishing my desired pronouns like this. How about XYZ instead"