I was at Disney between gall stone diagnosis and surgery four years ago. My doctor wasn't a lot of help with regard to diet - he just said to avoid fatty food until I could get the gallbladder removed. His main advice was that if a liquid wasn't transparent, I shouldn't eat it. So creamy salad dressing, for instance, would be a bad idea, but a vinaigrette might be OK. I just didn't eat salad dressing during that time. It wasn't worth the risk.
We were staying at Shades of Green, so our dining options were a little different. Unfortunately I lived on the kids turkey sandwich, which either has no mayo or could be made without mayo, and the spaghetti and marinara sauce at the restaurant at Shades of Green. If there was cheese on the sandwich (I can't remember) I would have pulled it off. My worst trigger foods were hamburgers and ice cream. Our only table service meal was Ohanas for Breakfast, so all the meat was out (sausage and bacon didn't seem like a good idea, though I don't remember what meat they offer.) We were there for Thanksgiving so we went off-site to Perkins for that main meal (they served Turkey and all the trimmings). I don't think I ate much of anything. I also remember some kind of rice bowl at the quick service place at Polynesian that we would visit on our way back to Shades of Green.
Unfortunately, your best bet may be the kids meals because they tend to be plain/without sauce. The trip was totally worth it - my girls were just a year old and loved the place. I was determined to go even if I had to go in a wheelchair - which wasn't necessary. Had I had a gallstone attack in the park, though, it would have been disastrous. They hurt so much I couldn't walk until they passed.
By the way, the surgery to have the gallbladder out was nothing compared to the gall stone attacks I'd had in the weeks leading up to it. On the plus side, I lost about 15 pounds while on the "gallbladder diet" - my coworkers asked how I lost it. I told them that the fear of a gallstone attack prevented cheating on my diet. They decided it wasn't worth it.