Your neighbour sounds awful and you shouldn’t pay a penny.
Sorry this happened, I’d also be terribly upset, but...
Your dog was in a secured space and was entitled to be there. The cat got unlucky through entering the wrong place at the wrong time. It’s a hazard of having a cat.
Tbh, free roaming cats are really problematic. When I was a child, our rabbit got bitten in our garden by a cat. Don’t know whose cat, but luckily we heard the scuffle and could chase the cat away. It had a collar, so was a pet.
A cat we inherited some years later lost an eye and eventually had to be put down after fighting with another cat.
All the damage to wildlife they cause by being natural little predators.
The fact they crap all over other people’s gardens (neighbour’s cat - I’m looking at you).
I like cats, but free roaming cats are difficult pets to care for and to responsibly own. The owner doesn’t have a leg to stand on. Much like the cat it would seem 