When dd had her milk challenge, although it wasn`t the true blind challenge, the doctor started by telling dd that she had mixed applesauce with ice cream. She was going to do a tiny bit of ice cream in applesauce and slowly increase the % ice cream. In fact, the first dose was straight applesauce. Dd had "symptoms". The doctor then told her the truth that it had actually been straight applesauce. This was a wake up call to dd. We managed to move on to more of the challenge, and then dd did have some real symptoms and failed it. Don`t recall what they were. Maybe your doctor can try something like that, say the egg is mixed with something else, but give straight something else.
(ETA: I just remembered: she had lip swelling when she failed the challenge.)
I actually think your ds`s vague neck itchiness might be real. One time years ago before any message boards, I let dd have fries in a restaurant. I had no idea of fryers being used for more than one item. She said she just didn`t feel well and lay down. No real specific allergy symptoms. After about an hour she was fine. The not feeling well and lying down, then being fine, doesn`t really happen to dd. When I discovered about shared fryers, I decided it was a cross contamination reaction. It gives me the willies now to think I ever let her eat restaurant fries (allergic to milk, egg, peanuts at the time).