We tried it for DS12 when he was your sons age. We did it for about a year and he was very good about not eating stuff he knew he was not allowed. (ex. ice pop at a party). We did not go as far as eliminating the dairy and wheat. I did not find that it made a difference in his being hyper. He has outgrown that now. It did make me much more aware of all of the artificial colors and preservatives that does not need to be in our food. I still buy many of the items and use some of the recipes. The brownie recipe is so easy and I think so much better than the boxed stuff. Some people say it has worked for them. Good luck.
PS-sorry just re read your post we did not follow it while at Disney. I would think though that you probably could since the chefs there are used to dealing with many dietary request.