I've met a few cats and dogs that have been bred so people aren't allergic to them, they're lovely, but very expensive, and like everyone has said, you'd be supporting breeders.
Your allergy doesn't sound too bad, my mum has very severe allergies to animal fur and gets very ill if she's even in a house that an animal lives in, even if it isn't in the room. I'd research what you can do to minimise reactions, and also if there's any breeds who's fur would be less irritant, because there's a fair chance that if there are and you search properly, you'll be able to find a rescue that's right for you.
Your allergy doesn't sound too bad, my mum has very severe allergies to animal fur and gets very ill if she's even in a house that an animal lives in, even if it isn't in the room. I'd research what you can do to minimise reactions, and also if there's any breeds who's fur would be less irritant, because there's a fair chance that if there are and you search properly, you'll be able to find a rescue that's right for you.