Can a soldier sue a private college for discrimination, based on military occupation, while the school takes federal money?

I read your story, legally you don't have a leg to stand on. The school isn't required to continue to teach you if it isn't getting paid. I understand they didn't do their portion of the TA, but that doesn't absolve you from being responsible to pay for it. Most colleges will work with you while the GI Bill and TA issues get fixed, but they are not required to. Assistance programs are for you to utilize, not for the school.
You also don't have a right be a member of an alumni association. Any private organization can deny entry to anyone for any reason. People not liking you isn't something you can take up in a lawsuit either. I understand you're angry, but it's a waste of time. Take your money somewhere else, report your experience to the accreditation Council, your unit, and anyone else you can think of.

I'm not an attorney but I think you will have a difficult time finding a sympathetic judge. An existing, long established religious school with a consistently well applied doctrine will not be forced to accommodate your wearing of the uniform if it violates their religious beliefs. The first thing that will happen under cross-examination is that the school's attorney will probably produce a statement of conduct which was required to be signed stating that a condition of admission would be to adhere to the rules. They will ask if you were aware of this? you might say "no" but then they will produce a document that you signed testifying to your intended compliance. Just an old guy's opinion but you probably should have changed schools as soon as you became aware of a conflict.

Anybody can sue anybody for anything. Question is it worth it and will a court care? In general, discrimination suits relate to illegal forms of discrimination that are codified in Federal or State law. For the Feds, it's Title VII. Of that, the only remotely close thing would be "religion" based on some negative result discrimination based on your religion. So a good attorney would look at what the actual damages are, if any. Suits for no harm get tossed quick. Suits that relate to something the school does administratively under their student code of conduct typically are not under civil court jurisdiction unless some law was violated to your disfavor. Just because you're mad or feel put upon doesn't mean it's an actionable thing by a court. BTW being banned from alumni activities is pretty much a nothingburger in the grand scheme of things. So unless you have real deep pockets, good luck. Getting some action group to go to bat for you is problematical. The Uber Left doesn't care about MILs and the Uber Right won't want to bash the Mennonites. These groups are more interested in their agenda vs. caring a dime about you. Now if you keep getting spammed and junk mailed for donations after being banned and they refuse to stop, then the topic can shift to harassment. Again, who other than you would care? Taking Federal Money? That would relate to grants and the strings attached or not attached. Again another legal morass. Bottom line, the world isn't a fair place and most unfairness isn't a court thing of interest. In most cases, taking a deep breath and moving on with your life is the best option. Bashing the institution on review sites invites action on their part that will give you a lot of grief until resolved. And you get what for that?