The burning in the gut was something else, that was small intestine bacterial overgrowth (SIBO). It happens sometimes when your gut is no longer in good condition. You need to take antibiotics to get help with that. You might also have the bacteria helicobacter pylori which gives you stomach ulcers
I will tell it to you straight from someone that was a immovable hardcore vegan for many years, I did a decade vegan and another decade after that almost vegan (with some cheese).
You can continue being vegan and that's fine but imho you will never gain back your health while doing so, not as a celiac. My skin had become so thin after 10 years vegan that it was almost plastic film over bone on my nose, I have no doubt my intestines were in much the same condition. At the time I felt fine, I performed quite well in sports and didn't seem to age as much as my friends etc etc.
But it seems my gut was just too thin and leaking and generally damaged from gluten at the same time. I got to a point where I needed to stop eating grains and nuts if I was ever going to go forward with my health and cure my immune system and unfortunately it meant I had no option left but to call in foods I hadn't eaten in years and that meant meat. Now being as extreme vegan as I was (I couldn't be in the same building as meat was being cooked and 2 or 3 times in the 20 years I wasn't eating meat I accidental got served some in food which disgusted me enough to instantly throw up, so I know exactly where you are coming from. But for me as sick as I was and with a son on the way, the time was right. When the time was right I had no problem eating meat.
I'm sure your time will come when you get desperate enough I only for your sake hope you can do it before you develop some autoimmune disease. I still love the idea of veganism but my I've had to admit in the end that my genetic makeup is just not suited to it. My wife has no issues whatever with living long term vegan but we aren't all the same.
Now I'm eating a whole food type diet and trying to eat the meat as raw as I can deal with (lightly cooked or cured). When I was allergy tested the main issue was with yeasts and my immune condition (Kimura's disease) seems to be caused by a hypersensitivity to yeast which you can safely assume was caused by having a very permeable gut for a very long time. Very doubtful the same thing will happen to you since it's a one in several million chance anyone develops such a rare condition but you have the right formula going for it to happen.
Anyway have a good long think about it, but I would suggest you do at least think about it.
Better to just do the genetic test and if you have the genes just don't eat gluten, pretty black and white. It's not worth taking the chance if your genetically high risk. My first child is due soon and he wont be seeing any gluten until he's had the test, if he's inherited DQ2 from me he will never see gluten while he's under our care. My wifes of spanish and asian american decent and is tough as nails digestive wise and totally immune to eating grains goes so I'm hoping he gets those genes from her. I'm engilish / irish and it's all bad news for grains for me.
I just had stomach cramps and diarrhea whenever I ate gluten and later on I had really bad acid reflux and permanent bloating and just felt generally like I was dying, I honestly didn't feel like I was going to live more than a couple more months at the time I finally gave up eating gluten, I was pretty sick. I used to get swollen hands and pins and needles. You could be deficient in something else, once your gut stops absorbing food you get deficient in lots of things. I haven't had stomach cramps or diarrhea for a couple years now but for the last year or so I was having a persistent burning pain in the guts, it's hard to say exactly how much pain that was because I was used to it, but you really had to twist the skin on my arm A LOT to induce enough pain elsewhere to make me not notice the gut pain, so I'm figuring it was pretty bad. It didn't really bother me though because like I said, I was used to it. It's gone now though.