The thing about lying about a food allergy is that allergies are treated with very specific kitchen protocols. These protocols make it possible for people with serious and life-threatening allergies to eat out. Every time a chef sees the word "allergy," they have to assume the customer’s condition is life-threatening and that they must avoid cross-contamination. The cooks consult a printed breakdown of ingredients in each dish to make sure the allergen isn’t hiding out in a component. They either grab new cutting boards, knives, and tongs or sanitize the ones in use. Even how the plate is brought out has to follow a protocol.
But when chefs and servers have people tell them they have an allergy, but then eat the food of others at their table, or claim a gluten "allergy" then order a beer, it often leads restaurants to be less accommodating. The people who suffer when this happens aren't people with preferences, it's the people with the allergies.
I don't know that Disney will become less accommodating. But claiming allergy isn't a "little white lie" at most TS or fixed-menu eateries. Even if they bring a dessert from another kitchen, those protocols come into play unless they're just bringing a packaged rice krispie treat or something.
