Our son is a VERY picky eater and we've had great success at Disney restaurants, including several in the world showcase at Epcot. If you are worried about it, put in a dietary restriction, other, when you book your ADR online and they will send a chef out to talk with you. They can make things that aren't on the menu, including cheese pizza with no sauce, plain pasta with butter and parmesan cheese, grilled cheese, flatbreads, etc depending on the restaurant. It seems almost all have chicken nuggets if your child will eat these (mine won't).
I'd recommend Akershus in Norway if you are on the dining plan. It's a great value for one credit. Lots of princess interaction, an included photo package, and the grown up food is delicious. Our picky eater loved the hot crusty rolls with butter, cheese from the cold buffet, and the chef made him garlic bread and then a plain pasta dish. Wasn't a very nutritious meal, but he was happy, so we were all happy. They even brought him a separate chocolate mousse dessert with no raspberry sauce underneath.
If you are paying cash, Akershus is not such a great value. I'd try any of the other Epcot restaurants that appeal to you and ask them to make something suitable for your children.