He isn't at fault.
The responsibility is yours as a dog keeper to keep your dog under control.
Had he shot your dog then he would have had a defence to the charge of criminal damage on the basis that your dog was damaging his property. That didn't happen.
However, the reality is that the Dangerous Dogs Act 1991 doesn't apply because no person was ever at risk or felt reasonable apprehension of risk.
The Dogs Act 1871 could apply but there is quite a lot of case law that excludes things like this where is is just chasing other animals.
the only thing that concerns me is that you talk of paying some costs which would imply there was some damage. If there was then you would be liable for it.
Can I clarify anything for you?
Pay for what?