Do I have to include my pronouns in a course outline?

Instead of the legal aspects, I would like to address the professional and ethical aspects of the pronouns trend.

In my opinion, the purpose of announcing your pronouns is to indicate that you will address other people with respect. The message that is sent is: since I told you my pronouns, you know that if you tell me how you want to be addressed I will not disrespect that preference.

There is a secondary purpose. People whose pronouns are not obvious may feel stress because they are the only ones who have to announce their pronouns. If you announce your pronouns, you are reducing that stress.

Therefore, the professional way to behave is to announce your pronouns. It's respectful.

Requiring people to announce their pronouns is, however, unethical. Some people are not sure how they wish to be addressed. Other people may not feel safe discussing their preferences. Requiring these people to announce their pronouns is abusive. Not announcing your pronouns is not necessarily wrong or disrespectful.

While I don't know if it's technically illegal, I'd say that picking a fight like this would absolutely affect your academic career. Explaining this, I think, requires unpacking some of the social circumstances around these issues.

You say "I personally would rather to not to do so for social and religious reasons." Stating one's pronouns is sometimes used as a slightly coded way of expressing support for transgender and non-binary people. It's important to realize that stating one's pronouns has clear merits outside of this though. Even if you think it's obvious by the way you look (and perhaps you have a name strongly associated with one gender), it may not be obvious to students from different cultures than your own. Even if confusion is unlikely, it hardly hurts to make it plainly clear and referencable, so picking a fight over it really begs a bigger question.

From your question, it sounds to my cynical brain that you do not want anyone to think that you are supportive of transgender or non-binary people. Please feel free to clarify, but without further information this is what I (and quite possibly your professor, peers, and students) would cynically assume, given that many such people exist.

I'll be frank. Without more context, picking such a fight sounds rather extreme to me. You aren't being asked to affirm the existence of trans people, you aren't even being asked to use other people's preferred pronouns, you are just being asked to put down how you personally prefer to be referred to. So why the fight? It's hard for me to imagine an honest answer that isn't at least in part what I've described above. If you are fighting about something so inconsequential, it makes me wonder how you'd interact with any trans or non-binary people in your class (including those in the closet). If I came to the conclusion that you weren't capable of being respectful to all students (part of your job as a TA, IMO), I wouldn't want you as a TA.

Critics may argue that I'm being too cynical or assuming too much here, and indeed if I actually had to make any decisions here I'd definitely be following up and asking you much more specific questions and not relying on reading between the lines like this. No, I say all this in hopes that I can help you and others understand why your professor, or other people like me, may have concerns with you picking a fight like this. If my cynical discussion above doesn't describe you, then you should make this extra clear to avoid having whoever you talk to assume the worst.

Many universities in the United States have a faculty member who serves as the university ombuds, uh, person. Here is what my institution says about that office:

An Ombuds provides confidential and informal assistance in the resolution of university-related concerns, especially those not being addressed adequately through normal procedures. He or she is an independent person who attempts to consider all sides of an issue in an impartial and objective manner. An Ombuds cannot impose solutions but can help identify options and strategies for resolution.

If such a person is available in your institution, make an appointment to have a chat. Do it soon because your syllabus is likely due very soon. If not, try to find a senior faculty member in another department who will talk with you about your dilemma.

A person whose duties include resolution of concerns, particularly, can tell you the official requirements for a syllabus, which will answer your "legally" question. Such a person can give you advice on whether declining to do what your professor has asked will damage your career at that particular institution.