If you've not already thought to do this, it's a good idea to suggest that anyone in your party needing to avoid dairy look at the Disney official menus. Allergy-friendly menu items are listed at the bottom.
There may not be more than 1-2 indicated "safe for dairy allergy" for any given TS restaurant. And, unlike pre-covid, a number of them may say they can't modify other menu items. What I was told was that most items are now "pre-prepared" with seasonings and marinades. As a result, the restaurant can't fix a plain piece of chicken or fish; they have very few raw ingredients currently.
Looking over the kids allergy menus may be useful. Sometimes, that's how I had to order to get anything but steak and salad since kids menu items were often not marinated, seasoned or coated. They will fix an adult portion.
I must avoid both dairy and gluten which proved far more difficult than in the past. Trattoria al Forno and Topolino's chefs created marvelous dishes for me. Either might be one to try.