Many dogs will consider cats as prey, unless they have grown up together from being tiny. Likewise, many cats will behave like prey around dogs -unless they have grown up together.
I'm not 100% into introducing cats into a home where there is an adult dog. Not unless I know the full history of both, and know for a fact they have lived and been happy in mixed households before.
For your cats' sakes you will either have to choose between blocking the dog door, or fixing up tight perimeter fencing (wire) -tall with an overhang inwards at the top, and all overhanging tree branches cut.....round every inch of your property. But even then, there is a chance of a gate being left open....etc.
As for your dog, I don't know about introducing a tiny kitten at this stage. There well could be an accident. He is already treating the cat as prey.
He can't help it. Part of this is his genetic heritage. I don't blame your dog. But you will have to be careful.
Sounds like your dog definitely has some prey drive whether it be going after the cats or just chasing them.
My question is, why do you keep getting more cats if they have access to outside? One has a,ready been hit. I think maybe you should work with what you have and block off that doggie door.
With a lot of dogs, they are going to chase something that moves suddenly or run. Maybe try keeping the dog on leash so you have some control when the cats are around the dog.