We get gluten free burgers at Pecos Bills in Magic Kingdom. We have had quick service at Cosmic Rays and the Hot Dog Place, too. But for burgers, we go to Pecos and it's always worked out great. Try to go off hours if possible (just before or after main rush). And prepare for an additional 10 minute wait. You can see photos in the food link in sig of burgers from Pecos.
Because the toppings are all open air, you may want to request clean toppings from the back.