I have my doubts about them not wanting you to show because you might "beat them". That's extremely immature. I agree with Jacksmama, it might just be the daughter thinking that. How old is she?
But... if your trainer does say no, I don't think there's much you can do about it. It's her property, her trailer, her truck. What she says goes (unless you're hauling your horse yourself. If that's the case, you can do whatever you want). I can't imagine why your coach of all people wouldn't want you to show. Why don't you ask her about it?