Barking at neighbour's dog

[3teens2cats]

We have a 2 year old small dog who is, in our opinion, well socialised and well behaved. He generally responds to a command first time and great with other people and dogs both out and about and visitors to our house/garden. We have just one problem and it's causing friction with next door neighbour. They have a large dog and our dog barks at him at the fence. This makes neighbours cross because their dog barks back and gets really hyped up, they scream and shout and then everyone ends up stressed. We call him away and he stops right away but apparently this isn't good enough and we must stop him from doing it in the first place. They refuse to work with us on the issue, refuse to let the dogs meet on neutral territory or engage in any desensitisation exercises. If i notice the dog before he does then i will distract him but not always possible. I think their over the top reaction is really not helpful but equally understand it must be annoying that ours sets theirs off. Any advice?

It is unlikely even if the dogs meet and get to know each other that the barking will stop when they are out in the garden.

Many dogs love a good territorial bark with their mates.

Easy option is to block the view of the neighbours dog. - although your dog may still bark if they can smell or hear the dog.

Classical conditioning may work so every time your dog is aware the neighbours dog is in the garden give him the best treats ever. But this will have to be done for a long time and will also have to be reinforced when the neighbours dog is in the garden even when your dog stops barking at it.

I would work on the classical conditioning option but my pockets are always full of liver sausage and dog treats so this would be easy for me.

Shouting by the neighbours will not help at all but I have no idea how to neighbour train unfortunately

If you spot the neighbours dog then gets some really high value treats and go and ask your dog to sit close to the fence. Keep giving him treats and requesting sits, touch, waits etc if the other dog goes stop the treats. Keep his focus on you. It will take a long time but it might work eventually.