To tell the neighbours the cat isn’t my problem

IME if a cat or anything else gives a Lab a snack they will be followed round all day!!!

I edited my other post before you replied so I'll add it here.

I am simply saying that if you know your dog has a tendency to chase animals, and you know your neighbour hates the dog, then it would be sensible to supervise the dog whilst it is outside.

My dog goes out, has a wee/poo on the grass, a bit of a sniff then she lays down in the sun.

The neighbours cats come in, wee/scent up all my plants and garden structures, scratch at them, bury their poo in my pots / borders, stalk and try to catch the birds and occasionally the squirrels. One destroyed half my daughter's fairy garden because a field mouse ran through it.

Well behaved dogs do minimal damage to a garden.

As I've said in my other post, harming animals can be used as evidence of the dogs character if the neighbour then goes on to complain that the dog hurt or scared her.

I am simply saying that if you know your dog has a tendency to chase animals, and you know your neighbour hates the dog, then it would be sensible to supervise the dog whilst it is outside.

the Dangerous Dog's Act does not cover animals which enter your properly secured property. The only animals specifically mentioned in it are assistance animals.

you could TRY to get vet fees or the purchase price of the cat but I am betting you wouldn't get far if the cat was off your property when the incident occurred.

I was trying to advise around the law. Your OP mentions your neighbours concern that your dog could kill the cat. Surely you see people with no control over their dogs daily, doesn't mean they are good dog owners or wouldn't be on the wrong side of the law if their dog caused harm. Lots of people believe 'I can do what I like on my own property' but it isn't true.

nope. cats are not classed as vermin. They are classed as property.

Eh? Vermin? They're in the same class as dogs in law, considered as chatel or property.
Both probably love their pets so lets be a bit more sensitive, comments like this are unhelpful.

Not really, you don't need a solicitor to do civil cases not in the UK. Would be small claims court. Doesn't cost much at all.

this. Cats are classed as vermin. Not your problem. Neighbour needs to provide a litter tray and toys, not expect your garden to provide. If you have children, give them water guns to discourage it too

Why would the council be involved at all?

If your dog killed an animal then it wasn't in your control at the point. It was out of control.

In my experience labs are big babies that would bit know what to do if they caught a cat, and cats are more then a match for most dogs. A quick snack from a cat is usually more then enough to send the average dog running away. Your neighbours are being ridiculous. I say that as a cat and dog owner. My cat runs rings around my dogs! Obviously you wouldn't want a cat harmed (although I do think this is unlikely) but cats are not than capable of getting out of dogs way as well. They are your neighbours though so not good to fall out. I would be kind to them and say of course you will look to see if the cat is out but of course you can't check the whole garden.

It's your garden Their cat their problem. The council won't accept their complaint.

This is why teaching recall is so important (which I believe the OP said they are doing)
Both of my dogs chase cats if ever they get into our garden (very rare), as they are protecting (I assume) their territory.

One of my dogs is a Gundog breed and I'm very aware of his breeding.
If he sees a squirrel or rabbit he gives chase, but he’s trained to come back when recalled, which he does. He has gone to chase a cat that ran out in front of us once, but turned around and came back to me when shouted.
The Cat in the garden is a different matter though, because it’s the dog’s garden they bark, the same way if someone entered the garden they didn’t know, they bark. They don’t bark at people on a walk so not all ‘garden reactive’ behaviour is copied outside. (Which is a pity as my dog walks beautifully on lead when training in the garden, but can be a bit of a twat outside on his lead 🤣)

Regardless of a dog’s prey drive, teaching recall is a must, but most dogs will give chase to a cat in their garden.

Whilst that is true, if OPs dog had previously killed a cat and the owner had evidence. The OPs dog then escaped and barked or growled at a person (or the neighbour claimed it did) this could be used as evidence of previous incidents with the dog.

a dog is also regarded as property.....I don't see what your point is?

My cat is a twat and torments the neighbours dogs in their garden. The fault if he gets injured is entirely his. He knows the dogs are there and he does it on purpose.

Civil matters cost megabucks. It may be possible to take such a matter to court but you'd need to be both rich and prepared to lose.

I would love to see that complaint to the Council.
Cats aren't generally stupid and it will soon figure out that your garden is no longer safe. I have cats and love them to bits but I also have a dog and while he has no chance of ever catching a cat the local ones know our garden is off limits or they will get a thorough barking at.