You know you need to be gluten-free. Even a little will cause antibodies to start destroying your body, even if you don't feel it. Eating gluten-free is a rule. If you cannot follow this rule when you are not at home, you will not go anywhere without a parent to supervise you. This includes school functions, movies, etc. If she is not following the diet at school, you could try to talk to the counselors and maybe she must eat a home packed lunch in the office. Maybe you come up and watch her eat.
If she was smoking pot when she went to a friends or the movies or at lunch, you wouldn't think these things were too harsh. Eating gluten is probably more dangerous to a Celiac's health than pot is to a 13 year old ( no I don't think kids should smoke pot).
Even if she says she will follow the rule, you will need a way to ensure she is. You will need to show up and check on her randomly. Maybe a teacher or another parent who is there will check for you. When my son first started driving, I would see if his car was where he said he would be. I had other parents and neighbors tell me how his driving was.
Yes this is a lot of work for you. You chose to be a parent so this is your job.
Yes, it will embarrass her. So what? I bet you embarrass her once by sitting with her for school lunch and a party and she will follow the rules. You have to mean it and follow through or it's a waste of time.