Our kids are picky eaters as well (no tomato sauce, ketchup or sauces; however, they'll eat mac & cheese, plain pasta, chicken nuggets, fries, bread, plain rice, popcorn & plain hamburgers and steak) and we were able to survive 3 days in Hong Kong and 7 days in Tokyo last summer. Search and review the menus on-line to get a general idea of what's available in the parks. If not, ask to having something removed or modified. My wife was able to get the heart shaped cheese and sauce removed from the famed 'chopped' steak from Queen of Hearts in Fantasyland. The Manager was even concerned about any food allergies. It may have taken about 20 minutes for my daughter's lunch to arrive, but she was happy (I missed a great photo).
Before we sat down at a traditional Japanese restaurant at the Sky Tower, a local resident (Costco employee

), took me to Family Mart (7-11) to buy a package of white rice to which the clerk heated in a microwave for a 100yen ($0.85). My daughter had something to eat at a very reduced rate. Not to worry, we spent enough at that place and we just didn't waste any food on my daughter. To locate one of these marts, just look on Google Maps and you'll see their symbols everywhere.
For breakfast we ate at Krispy Kreme one morning and a Belgium Waffle shop next door on another day in Ikspiari. There are lots of eating establishment there as well as a small grocery store with good prices.
Don't worry as there are lots of choices within the parks and outside as well. We even eat at sushi go round in Shibuya (west side of Tokyo) and the kids ate baked chicken wings, rice, small hamburger patties and cooked shrimp. Sushi was about 75-80% of the menu, but they found something to eat. Maybe the food being sent via trolley was a fun distraction.
One last thing, portions are not as large as in the US, but we were full and satisfied with the variety and flavor of all of our meals, Western or Eastern (we even lost a little weight).
Again, Check the menus available on-line to get a general idea of the choices available and ask the clerk/server to slightly modify any food items to your needs. They were very helpful and we had no complaints. Best of Luck and Enjoy!