All of the sit-down restaurants at WDW will accomodate special dietary needs, especially if you can give them 24 hour advanced notice. In my experience, even the fixed-price meals, such a buffet service places in Epcot like Akershus in Norway, will make exceptions for people with special needs if you ask nicely. I take my son's food to restaurants at WDW all the time, and have never had a problem. Some character meals will charge an infant charge or entertainment charge, but never full price for him since he is not eating the food there.
I am not sure how you will fare trying to order from a children's menu, they may not accept that, but my guess is that they will be very accomodating to your requests. Whether or not you have made a reservation with a special dietary request, be sure that you inform both the host and the waitperson that you have a special dietary need.