I don't think that there is a lot of leeway at CS places. We spoke with a specialist at Disney that handles food restriction questions prior to our trip because of my nephew's allergies: shellfish--extremely serious, could be fatal, and eggs--not as serious, but still needs to be avoided.
She told us to do our best to avoid eating at the counter service places because most of the food is assembled/prepared elsewhere on property, and they cannot be absolutely certain of each ingredient. For example, she could not guarantee that there would not be egg in the batter at the walk up window at Rose and Crown, but Raglan Road was able to provide egg free fish and chips.
If you need things made to order, you would be better off at TS places for the most part.