I dont have much help for you, but I do sympathise, because I deal with it EVERYday. Bailey is bad when the cat it coming in from outside he hides under the table, then attacks, or when I'm trying to leave the cat out. If I try to pick the cat up, to put him out, he bark and jumps and will grab at the cat ( usually getting a wad of fur) OR he'll bite at me! Which is really bad,and if I yell or say NO, he is so into the attack he doesn't even hear me.
I have tried a water bottle ,, squirting Bailey while the cat is coming or going.
See I'm no help, but I know what you're going through.
I do block of the hall and spare rooms with a gate, so the cat races to that point and jumps the gate,and Bailey can't get to him, sometimes the cat sits on the otherside of the gate and swats at him through the holes.