There's another way that the protein can affect allergies though- if it really is hard on the body to digest so much protein, the body will be working harder at that and will lack the resources and energy to fight off infections or tolerate allergens. They might not be allergic to the meat protein itself, but the weakening of the system might affect other systems.
HO, you were being simplistic about the protein right?

Because even if you're allergic to say, eggs, you might only be allergic to one type of protein in the eggs, when there are many, many different proteins in an egg.