I did Disney in September on a ketogenic diet. (Mostly. When I cheat, I don't get seizures, I just get severe facial pain that is partially controlled with drugs, so I did cheat 3 days. . .and paid for it, but it was worth it for me.)
Okay, the QS restaurants mostly suck except for breakfast. For lunch, I got double burger with bacon specials and ate the meat out of them then tossed everything else. For breakfast though, they're pretty good. We were at Jambo house, and I got eggs and a meat as my breakfast every morning.
For ADR places. . .
The Sci Fi Dine In at HS actually had something on their menu that I could order with NO changes!

They have a blue cheese salad that's just a wedge of iceburg lettuce and it comes with steak and NOTHING else! (Okay, I think I did ask for no tomatoes, but that was the extent of it!) W00T!
At any of the buffets it would be v. easy to stick with all protein/fat in the right combo and stay under carb limits. We did Jiko and I had no problems both stuffing myself and staying under my limit for the day.
We did the Marrakesh and I believe I got a kabob with nothing else on the plate and mussels as an appetizer. Stuffed myself quite handily! (ETA: DH thinks I got veggies with my kabob and just picked through them and ate what I could, and he might be right. I remember mussels and the kabob. . .)
Based on that, you should be able to manage, although at the QS places, I think you'll be tossing a LOT of food away.
We ate at the Cracker Barrel off property twice though, because I can get a 5 carb meal there very easily. I'm an adult though, and so long as I stay under about 20 carbs a day, I can stay in ketosis pretty easily, so I have a teensy bit of wiggle room but can still keep the stick all reddish
ETA: I never even thought about calling the special foods department - might do that next time. I did check all the menus out on this website just to have a sort of game plan, and DH and I both have nutrition facts websites bookmarked on our phones so we can count carbs on the fly.
ETA2: I forgot to mention - we went to the Target as soon as we got to Disney and I bought a bunch of cheese sticks and pre-cooked meats and such for snacks that I could stick in my pockets for noshing purposes and for bedtime snacks.