On our last trip to WDW, our D, who has food allergies, had to continue with an elimination diet her doctors had her on, and she could have no dairy, eggs, or soy. She was already a vegetarian, so this essentially meant she now had to eat soy-free vegan. When we made our TS ADRs, we noted on the reservation the allergies to egg, dairy, and soy. When we showed up at each ADR, we were given pre-printed allergy menus, where it would list items that were safe to eat for people with certain allergies. Shortly thereafter, the chef came to our table and discussed our D's food restrictions and what she'd like to eat. Where possible, my D ordered something off the allergy menu, but because of the way the allergies are listed and combined, this wasn't always possible. Also, sometimes my D didn't like any of the choices on the pre-printed menu, and the chefs were very happy to accommodate her and make her something she'd like.
In particular, we had fantastic experiences at The Wave, Mama Melrose, and Trail's End. At each of these places the chefs were absolutely wonderful and made her the most incredible meals that other diners were looking at jealously! In particular, the chef at MM made us all the most delicious roasted veggie pasta we've ever had in our lives (we all followed her diet because she was really upset and we didn't want to rub it in by eating all the stuff she couldn't right in front of her). He asked her what her favorite veggies were and she mentioned eggplant, sundried tomatoes, mushrooms, and red pepper, and he just went to town making this incredible pasta for her/us. And Chef TJ at Trail's End is legendary for making spectacular meals for vegans and people with food allergies...he made us/our D a 7 course meal that was seriously the highlight of the whole trip for us! So I really think you will have no problem at all at TS restaurants. The Disney Food Blog has a video called "Best WDW restaurants for vegetarians and vegans" that you might want to check out. Just be sure to plan out these TS meals in advance and make ADRs so there's no need to scramble while you're in the parks.
In terms of CS, that gets a little trickier, but it can be done too. All of the CS locations have binders with ingredient lists in them. What I did for our trip was to google "vegan disney world" (which would work for you too with your child's allergies), and I found several websites that listed specifically which items at each park were vegan (for example, all popcorn and Mickey pretzels, minus the cheese sauce, are vegan). I then created a Pinterest page for each park and pinned photos of the specific items that were safe for my D to eat -- we found that much easier to do than follow a list. That way, you can access your Pinterest page on your phone and will have a quick and easy guide to CS meals...and I found that the photos helped our D to be more excited about her choices.
And again, I do appreciate your child isn't vegan...but the dairy/egg/beef allergies make it close...so it seems like there would be some options to add in grilled chicken to, say, an otherwise vegan pasta or salad, and that might work...or to add a vegan bun to a grilled chicken sandwich.
Good luck! I hope you have a magical trip...and I really do think Disney will take great care of your child...we found it easier for my D to eat there than anywhere else we've ever eaten.
