I agree that most Disney restaurants can accomodate vegetarians, and most can accomodate vegans. Many can accomodate pescatarians, and all try to assist with food allergies. This is none of those. This is a picky eater whose Mom wants to eat at LTT, and will place the manager on the spot when they check in.
My DGD had several food allergies when she was younger, and her pediatrician was strict in regards to what she was allowed to eat. DD spent a great deal of time on the DIsabilities forum so she would be prepared while Disney Dining. We knew that there would be times when DGD Might not find a meal in the menu, and the chef would not be able to help. Thankfully DD chose restaurants that had items she could eat, so we knew that even if she did not have all the sides, she could have a Protein and often something else. But DD also had food JIC. The issue was theirs, not Disney's.
I think that there is a huge difference between preferences, like the OP's son and food allergies or those who make conscientious decisions in regards to eating meat, etc. The restauant the OP is discussing is an AYCTE meal, that has specific items included. The child cannot opt out and have just part at a reduced cost, so unless there are items the kitchen can readily make, like veg on the menu, and salad, it is kind of rude to ask the manager to accomodate, IMO. I would book someplace else.