Re: My three year old daughter ...
You mention all the foods that your daughter eats that you feel have no connection to the hives but have you looked at common ingredients? Things that are in most or all of the foods she is eating. Especially CORN. Corn is in everything. Corn syrup & High fructose Corn syrup are obvious but look out for corn starch, corn meal (on the bottom of breads) corn flour, dextrose and any other by product of corn.
My wife suffers from hives and while hospitalized for a really bad outbreak they almost killed her with an IV that contained some product of corn, I think it was dextrose but I really can't remember. Her entire back swelled up into one giant hive.
We only figured out the connection after extensive allergy testing by a second (competent) allergy clinic. The first one was an idiot! She is allergic to Corn & Apples.
Apples have been fairly easy to avoid though you have to watch for it in the form of pectin and in any juice product that has blended fruit juices.
She still has hives though they are less prevelant than before we made the connection to corn. When she breaks the cycle she will still have a few small hives but they come and go quickly. They return with a vengenance when she accidentally ingest anything containing corn and it happens within minutes.
Besides allergic reactions to corn I would avoid it just because of all the genetic modifications they have made to it.
Anywhere you go to eat request the ingredients list of what ever you are ordering for her and if they refuse to give them to you then just leave and eat somewhere else. We have only had one idiot refuse, everyone else has been really helpful.
I hope you find a cure for your daughter.