Alla:
I think you'll find that the Disney chefs are very accommodating when it comes to guests with allergies. Make sure to let them know when you are making your reservations. The chef will actually come to your table and discuss your preferences. With that in mind, I'd recommend a table service restaurant with menus versus a buffet with set offerings.
If you will be in the Magic Kingdom area and want something fun, consider
Whispering Canyon Cafe at Wilderness Lodge. It's just a boat ride away and would appeal to a youngster with its theming and stick pony "rides."
California Grill at The Contemporary is another option and is the first monorail stop after the Magic Kingdom. The tab will be more in line with Cinderella's Royal Table, but the view is memorable. My meals here have been fabulous and with their varied menu, I don't think you'd have any trouble accommodating your daughter.
The Wave is also located at the Contemporary. It touts its fresh offerings, but it would be very bland surroundings for your child.
Lastly, if you don't want to leave the Magic Kingdom, both The Plaza Restaurant and Tony's Town Square Restaurant are menu-service restaurants in the park. The Plaza offers upscale salads and sandwiches while Tony's is Italian. And don't forget that your best bet may be simply asking to see the "allergy book" at any counter service restaurant like my favorite, Cosmic Ray's. They keep it behind the counter with the ingredients to everything they serve.
Wishing y'all a magical vacation!
Whitney