I have a cat and a dog.
The dog is absolutely nuts in the best way but a total wuss, he runs from butterflies! 🥴
He adores my cat and the cat adores him so he's used to cats however I know the risk of my cat being an outdoor one, he's barely in (has a cat flap) apart from food and on a rainy or windy night.
I know that he could cross paths with a fox, jump into somebody's garden who's dog could do this or be hit by a car.
It is not your fault, your garden is secure, your dog was on your property, this is the risk outdoor cat owners take.
Offering to pay half was generous of you, as for him being rude and threatening to prosecute, let him do what he wants. You won't be in trouble unless your dog attacked whilst you were out (dog out of control).
People keep guard dogs in their gardens fgs.
I feel sorry for their cat being hurt but their insurance will cover the cost (unless they don't have any which is their problem).
Hope you're okay this morning cx