The dove will not recover, especially in a carrier or cage. It’s in shock. Place it on your garage roof or something. It has a mate who is looking for it. That may help. If the cat punctured the skin anywhere, it will die.
Keep the cat in the house! Is there any bird rescue’s in the area? Doves do have mates.
There is no blood anywhere, and he’s resting comfortable. He can’t fly. Turning him loose would be a death sentence. I’ve read rabbit pens work, but I don’t have one.