The tests they do for celiac look for full-blown celiac. The allergy tests look for full-blown allergies. So if you have simply an insensitivity you will not get any results from any tests at the doctor's office. The best you can do is what Leida mentioned. Do an elimination and see if there's a difference.
I do not have any issues with wheat. I could eat some right now and not feel a thing. However, I never knew how much better it was possible to feel until I eliminated wheat and other grains. The difference is like night and day. I feel young for the first time in my life. I feel like I should have felt when I was 17 years old, only I'm 47 now. I feel like leaping and running and jumping for joy all the time. My life before was like swimming through molasses.
Female, 5'3", 49, Starting weight: 163lbs. Current weight: 135 (more or less).
I can squat 180lbs, press 72.5lbs and deadlift 185lbs