Need help from the pros

I originally thought this was a feral cat. He showed up one day and a month later he had not left. The baby rabbit population was decimated. I tried calling to him one day and he ran the other way stopping briefly to hiss over his shoulder. I put food out and he ate. I put more food out and he let me pet him. I put more food out and he bit me. I put more food out and now he falls asleep in my lap. My problem is twofold.
1. I own two dogs who are not youngsters and have never lived with a cat. One of the dogs is intrigued by the cat and just wants to know him.
2. The other dog is my bestest buddy in the world. He has no interest in the cat but he is going through living hell health wise. I do not want to disrupt the remaining months he has to live.
3. My Significant Other said if a cat moves in, he moves out, but now he refers to the cat as "our cat"
4. I know my S/O will want the cat to be indoors when the weather gets cold. He's a funny guy;)
5. I have been trying to introduce them to each other slowly. I got Casey the dog and Osee the cat wihin 6 inches of each other tonite, but I would fear for any of their well being if left alone for more than 2 minutes.
6. What should I do???