@Bluetoybear OP, I haven’t seen someone mention this but maybe I missed a comment. Have you checked your dog over very, very carefully? With fur, you can sometimes miss scratches or puncture marks if the cat scratched or bit your dog. The fact is, unless you saw what happened, you have no idea, so you don’t know your dog is responsible.
For all we know, the cat got too close, the dog barked, the cat took offense so swiped him in the nose with claws, so the dog thought it was defending itself by kicking off. In fact, you don’t even know for sure that the cat wasn’t injured when it jumped into your garden (it’s possible - if it’s never entered your garden before, it could have been struggling to get home) and all your dog did was get very upset at its presence, chase it, and possibly nip at it. I’m more of a cat person than a dog person, usually, but your neighbors are being complete arseholes.
Oh, and I’m sure there’s no way Mr. Neighbor would have yelled at you or tried to knock on your door forever if your husband had been home. What a bullying prick. Don’t give him two pence.