Honestly, I think you mother has half a point. Hear me out.
It doesn't sound like he takes his condition completely seriously. I used to work in a restaurant also, but I was getting glutened constantly. I made the decision to move into a new line of work after a sufficient amount of time passed when I was being extremely careful and still getting sick. He needs to face up to the fact that he is going to have to find another job. If he isn't willing to do everything to make himself better, then your mother has a point.
While accidents happen, if it's happening more than once every 6 months or so, then he probably isn't being careful enough.
I do not think it would be wise to get married to someone who was not willing to take their condition seriously and take 100% responsibility for it. Whether or not he has a disease is not your mothers business. Whether he has a disease and he is not willing to follow the treatment fully, knowing it will hurt both him and you if he doesn't, that is her business.
I agree with you about his working in a restaurant. Unless he is in a gluten free restaurant and it doesn't sound like he is...he's not taking this seriously enough.