We have 4 cats and 3 dogs. Our husky has high prey drive and has nearly killed 2 of our cats. If he could get to them, he would kill all 4. We have a 2-story house, so the dogs live downstairs with access to the outside, and the cats live upstairs exclusively. So both the dogs and cats have an entire floor to themselves. It's plenty of room. The cat's food and litter boxes are upstairs with them, so they have no reason to EVER come downstairs. They also know that Paw Paw is waiting to eat them, so they're smart enough to stay upstairs. All we have up to separate them are baby gates that the dogs don't knock down or jump over, even though they could if they wanted to. Only by accident has Paw Paw gotten to the cats, and that was because our children accidentally let him upstairs and my wife or I were on him in a heartbeat. We also have a door we can close at the bottom of the stairway landing. Not sure if you have the same option, but no cat or dog can get through a closed, secured door. Haha!
I honestly don't think you can train prey drive out of a dog, but I'm no expert. It seems like a natural instinct that's there for a reason. Not sure if your dog has prey drive like my Paw Paw. He'll catch birds out of the sky. He's fast and intense! I've never seen a dog like him. But we keep him under control, because we love our cats (almost) as much as we love our dogs. Haha!
Good luck. And I'm there with you about idiotic cats. When we had the big wire dog crate still out for overnight sleeping/training for our pup Beowulf, our rescued feral cat somehow was running from the dogs and ran into the crate's open door and got herself trapped inside with all 3 dogs attacking her. They must've gotten 30+ bites on her before I could wrestle them all out and save her. Luckily for her this time, there was not a scratch and she was fine. Whew! I'm definitely a dog lover, and these cats have a soft spot in my heart, but I'd be OK with just dogs and no cats. My wife would disagree, thus a compromise is in place.