Sounds like a nice cat, and the dogs will probably like having a new friend to play with. Try putting the cat in a pet carrier, etc. to introduce it to the dogs. If that goes well, slowly introduce them from there – maybe one dog in the room at a time to start.
The claws are no big thing. At worst the dogs will maybe get a swipe at their nose “to teach them who’s boss” if they are aggressive with the cat. If they all become friends, cats will just give them a no-claws-out slap when they’re playing. If you’re still that worried about the claws, just use a nail clipper to blunt them slightly.
If you do keep it, you MUST feed it CAT FOOD. Cats that eat exclusively dog food can go blind because dog food doesn’t contain enough taurine – an essential nutrient for cats.