A dog comes into your home bites your dog and is growling at children? You try to scare him off and he doesn’t leave. I am not sure what you are not understanding here.
I'm not against shooting a dog attacking someone and I am not at all trusting of pits or pit crosses. I would do the same thing given the right circumstances, but something about the story doesn't sound quite right. The dog is outside, in the driveway, your family is safe inside...pick up the phone in one hand and call animal control and put your gun in your other hand and wait.
Re-read the story again. you’ll see why.