Start with water only as a beverage. Hydrate but only with WATER. No Mickey Bars, no icee drinks, etc. Don't pack any hand-held snacks or toddler food. Or IF you do, make sure they do not know it's there. At actual mealtime, tell them they can have the light-up sparkly kids cup AFTER they finish a palm-sized amount of food. That is the size of their stomach and we as adults often WAY overdo it on portions. And sooooooooooo much liquid and treats are empty calories.
They might be hungrier than they think after being stimulated, hot and watching other people eat delicious-looking and yummy-smelling food. The scent of popcorn never stops so it's really easy to entice them into popping a snack into their mouths. Just remember you are the parent and you decide WHEN and WHAT they are served, but the child controls HOW MUCH and WHAT they eat.
If they pitch a fit over not eating the entree, you just pack up the sides (most are pre-packaged already) and remove your child from the eating area. And they don't get to wail over wanting an ice cream until they eat the apple slices.
My mom, fully intent on ensuring NO HEATSTROKE in January and hydrating everyone, gave my DD5 a 1liter enhanced water/energy drink thing as we passed Security. Gee I wonder why we all had a Family Meltdown when by 10am we had only done FIVE potty stops, no rides, and two gift shops (because kids who aren't whining for plastic crap totally NEED it) and only crossed off two M&G. After Grandma got fed up with our constant requesting NOT to spend all day shopping and hiding under like the 4 trees in EPCOT while sucking down this massive beverage... she left the Park and then we were in a struggle for the rest of the day with DD to eat something, anything . A whole meal at Garden Grill = WASTED, PLUS a cancelled Coral Reef ADR, because of some stupid pink carbonated beverage. Not to mention NOT eating and drinking around the World Showcase during the very first International Arts Festival!!!! we were on the Dining Plan and I think we used like 4 credits that day, it was so bad.
Some places can make substitutions but I think it depends on their kitchen. So you have to really be the flexible one IF the diet choices are not due to medical need.