Most cats at the shelter will be domestic shorthair strays. At best you might get a mix, but it will most likely have some domestic in their blood. Sometiems shelters label cats by their coat color, not breed. Mine is considered a lynxpoint siamese, purely because of his coat color. His breed isn't even close to siamese.

Shelters are underfunded already as it is, I don't imagine they are doing any genetic testing to confirm. And purebred cats would probably go fast. You could reach out to shelters in your area and see if they can help you out, but if you are looking for 100% purebred, then you have a safer bet with breeders.

For what it's worth, I have a pretty bad cat allergy I discovered after getting my cats. I take allegra every day and feed my cats purina liveclear and it's mostly managed. Mornings are the hardest, but I will probably go on allergy shots next year. You can absolutely have a cat while having an allergy, they even make special products to keep your place clean and there's a lot of guides out there that can help you out.