i have a long list of common foods that i need to avoid (wheat, dairy, soy, rice, potato, tomato, etc.) and so it is sometimes difficult for me to find food to eat when we're not in a full service restaurant.
the woman i spoke with at the chef's office for MK told me it would be fine to carry in what i needed. no entrance guard ever said a word about the small soft-sided cooler i carried in, which had carrots, applesauce, hummus, etc., nor the bag of taro chips which was outside the cooler. and i can only imagine that a 2-year-old would eat even less than a 40-year-old!
i think it might be disney's policy for people to not bring food into the parks, but i do not think it is enforced, even for people who do not have special needs. a friend of mine tells me he brings in sandwiches for his family all the time to save money.
i really don't think you're going to have any problem bringing his food in.
have a magical trip!
-dj