At Disney, they have access to a book with a copy of what each state's driver's licenses should look like. If the non driver id card looks similar, I think you're fine. If it's very different, especially if it doesn't have a security feature like the holograms, you might have a problem. If you have a passport, it is pretty difficult for someone to call that a fake! Maybe a copy of a birth certificate? Maybe you could call Disney and ask them what the best form of id in your case would be?
It's funny, in my area, a permit is just a piece of paper and a non driver id card looks almost exactly like a license!
Good luck!