spod it sounds like your consultant is a paediatrician with an interest in allergy. Any chance you could be referred to an allergy consultant? Although some doctors with an interest are very good they aren't IMHO, quite the same, especially over things like age at which they will test.
You could try keeping her off egg but then trying gluten. I'd give her probiotics daily for two weeks first (and while you do the trial). Oats cause fewest problems and should be tried first. If you try those and she is ill don't go on to anything else!
From what you describe I suspect the problem is not gluten as such (so she will improve with age) but a general gut problem and possibly mild dehydration.