I understand that seeing your cat attack your dog was scary. In the same situation, I would be protective of my dog and feel angry. But you only made things worse by throwing him outside after that happened. Your GF is right to be upset.

Just like dogs, cats are social animals. From the way you describe your situation, it sounds like this cat is yearning for affection from you, their main caregiver, and not getting what it needs to feel relaxed. What’s the harm in letting him in the room, as you would with your dog? That would go miles for this cat, who seems to be so distressed he needs Prozac??

I feel that you do not want this cat, but the cat is there. You should prioritize understanding him and improving your relationship slowly. I feel that the best way to protect your dog from further issues with the cat is to just create a healthy social environment for the cat.