I know this post is a couple weeks old now, but since my experience has been quite different than the other posters I feel I should share.
Among my many allergies, I react anaphylacticly to fish. I also have allergies to those not among the top 8.
My worst experience was at the GG in EPCOT. The chef was amazing. He even prepared parts of the meal the day before and was very thorough and knowledgeable/willing-to-learn about my food allergies. Since it was a character meal, my family was caught up in the moment with interacting with characters when our food came. The chef brought it out to me and it seemed fine at first. More characters came and went and by that time I was very eager to eat. As I was cutting up the first meat on my plate, I noticed that the meat was fish. The chef apologized, but I assume that an underling back in the kitchen was working on auto-pilot and put fish on my plate since the GG meals are all the same. If I had not known what fish meat looked like, or if I had not been paying attention to what I was eating, I could have died or at least the rest of my vacation ruined. I am scared to think what would have happened if I had been a child and had eaten the whole piece of fish unknowingly.
Your doctor is right in advising you to avoid places with the potential for cross-contamination. People on the boards will say that Disney is great, but that is only their opinion based upon their positive experiences. I have had MANY negative experiences at WDW and have discovered numerous flaws in their system including unknowledgeable/uncaring/lying CMs and chefs to unlisted ingredients in meals (BOG restaurant with the new computer system). The reality is that many working in the kitchen have almost no knowledge of food allergies. They are either young college students there for three months or older folks who think food allergies are a modern fad and don't take them seriously. These guys are minimum-wage workers. Do you really expect them to understand the finer points of cross-contamination? Are you willing to trust your life with them? Your child's life?
Also, I would suggest researching about fraud within the seafood business. Much of the seafood around the world is contaminated by fraudulent practices of selling a cheaper look-a-like product in place of something more expensive. For this reason, I don't eat any seafood because unless I know that the food came right off a boat into the kitchen, there is really no way to know if there was a fraudulent substitution. This also commonly happens in the olive oil and honey industries and affects those with corn, sunflower, and soy allergies.
Sorry to be the bearer of bad experiences and warnings.