My son has had celiac disease all his life, from a long time before it became trendy and it'll last till the end of his life a long time after its off the front page. He's one of the 1 in 100. This was before wikipedia. Perhaps a future dating system will number years as in BW before wiki and AW after wiki. Anyway needless to say I recall quite clearly, his pediatrician FINALLY did a blood test as a last resort (celiac is not a profitable disease to treat, therefore rule out everything more profitable first) which had wildly high results, then refer to gastroenterologist who knocked him out and did an endoscopy, "yup all inflamed and Fed up in there" then trial run gluten free diet and his blood antibody numbers started dropping and he nearly instantly had a normal digestive system. And like yourself he gets horrifically sick a couple hours after eating gluten containing foods and it lasts and closely resembles a very, very bad food poisoning incident (like to the point of dyhydration danger, and yes he has been poked with IVs in the ER for this). So I donno about all this "there's no tests" this is extremely heavily trodden ground. Your primary care doc might be pretty ignorant of this specific disease but I can't imagine a gastroenterologist being confused, this is after all kinda their bread -n- butter...
On the other hand I donno if you technically "need" a medical diagnosis. I'll spare you the details, but shellfish and plain ole chicken eggs don't seem to be terribly well tolerated by my digestive system... solution? I don't eat them. I don't need a trendy diet or diagnosis or a pill or celebrity endorsement, I just don't eat lobster or fried eggs.
Note that wheat provides approximately nothing valuable to your body other than empty calories, so feel free not to eat it. Also almost all junk/highly processed foods are stuffed with wheat as a filler so your health will most likely dramatically improve if you stop eating wheat. The reason trendy actresses feel better after stopping wheat is not the lack of wholegrain bread but the lack of twinkies, not the lack of soy sauce but the lack of an entire pan of lasagna in one sitting, etc. Its not at all like giving up citrus or giving up meat where you have to put at least a minimal amount of care into making sure you don't end up with a major nutritional problem.
BTW his whole digestive system got messed up and he's also allergic to soy and milk casein proteins. Supposedly if you "tough it out" for a couple decades you can get cancer from the constant inflamation. Also his growth was pretty stunted until he got a proper diet (at which point he started growing so fast I swear you could watch him get taller). So get this checked out. I may not be a medical doctor, but it doesn't take a PHD to figure out if it makes you sick, don't eat it.