I think Disney overall does a great job accommodating allergies. However....it is possible to go to the place people thing is the best and run into a chef who does a terrible job. I think when you have allergies to more than one thing it increases the chances that you will have problems - or at least fewer choices.
Since your child is so young it might help if you suggest to the chef what items your child really likes (hot dogs, chicken nuggets, pasta, rice, mac and cheese, pizza, grilled chicken, veggies, etc.) and see if they have a safe version and they will make sure the restaurant has it. I would especially ask about the
desserts if that is an issue (my son with dietary issues used to dislike any ice cream and often the only safe dessert was tofutti - which he didn't like. So it made it difficult for the rest of us to have dessert - like we could choose from the whole buffet of desserts and he had only one choice and it wasn't something he liked).
Look at the menus for the restaurants online so you know what they typically have and what might be safe or they could do a safe version of. The menus can be found at allearsnet.com and here too I think.
I think the biggest problem is that with many restaurants (especially those in the parks) you end up talking to someone at a central number who passes the info along to the chef. You don't actually talk to the chef directly - that did not used to be the case. At some places they still give you the # to the restaurant and one of the chefs will call you. Yet it might *not* be the chef who ends up working the day you are there and preparing your food.
Brenda Bennett is very helpful and you definitely want to give her office a call and they will email you lists of items at CS and carts that may and may not be safe.
Chef TJ at Boma is probably the best on the whole property - but I can only recommend Boma IF you go on a night he is there. I think you'd have to make your ADR (maybe make a back-up at the same time for another place though I hate to say do two but sometimes you gotta) - call close to the time you are going and find out if he is working that night and if he can call you to discuss.
We've had good luck at Crystal Palace, Le Cellier, Biergarten, Mama Melrose, and the Rose and Crown. Chef Mickey's was "ok" though some have gotten great service there (I had a terrible chef - luck of the draw).
We haven't done the princess meals cause I have boys - except for the one at the GF (with the stepsisters). The food there was so-so but it was fine - We got a big kick out of the characters though! We have eaten at CRT and Norway without the princesses and both were fine but that was a few years ago.
Don't worry too much about it. I think the older the person is and the more sophisticated their tastes (and adult vs. a 4 year old will mean a VERY different meal usually!) makes a difference too.
My younger son is allergic to peanut and tree nuts and a type 1 diabetic - fyi. Don't worry!
DVCDawn
Yak & Yeti. They didn't do "the routine" like the other table services. The CMs in front did not pull us up on the computer (which everyone else did, and therefore saw the allergy note) The chef didn't seem to know we were coming. He didn't have the notepad. There weren't as many options. It seemed more like they weren't expecting us, even though they were notified far in advance and all the other table services were prepared.
