Let's see here:
1- Check the traditional Iranian bakeries in the areas found mostly in the co-op squares (khabaz irani). For one, I'm sure they don't use dairies, and I'm pretty sure you can provide them your preferred ingredients and ask them to bake your bread using it. Of course expect to pay more than what you usually pay for pre-baked bread bought at stores.
2- Have you tried the specialty supermarket and groceries? Saveco, The approved Market, Lulu Hyper, and there's another one in Salmiya behind Marks & Spencer that I forgot its name (part of Saveco).
3- Rumaithiya Iranian bakery in Block 7 is very different than the regular ones, they bake their break with no yeast. Check them out 29.316279, 48.078299
4- I know I've seen at least one Instagram account of a home-business that specializes in baking special bread for different diets. Here's one I just found @figsoliveskw.
5- This should've been #1, but Nature Land stores are super healthy and organic in everything that comes to mind. There's a store in Qadsiya and I've seen many types of dietary bread. If you didn't find baked bread, you can definitely buy their dietary flour (wheat, gluten free..ect.).
This is what came to mind so far, I hope you find it helpful and get to enjoy bread. Cheers!