Some people have simple wheat allergy without celiac.
I happen to not be able to eat bakery products, but not because of wheat or celiac. I'm allergic/intolerant/sensitive to the "dough conditioner" - sulfite. Bakeries mostly don't mix their own ingredients. They buy everything, bread, cakes, rolls, everything frozen. Pizza crust too.
I get gastric reflux which triggers tachycardia.
No, it's not sulfate or sulfa. Just SO3- sulfite.