He expected me to call him to only safely retrieve dogs during / after they attacked the chickens.
I do not know if this would be safe, or even if it would be possible in your area, but maybe live-trap the dogs on your property and take them to the animal shelter? Some shelters charge money before owners are allowed to retrieve their dogs, and any shelter would require the owner to go there to get the dogs back. If the dog is not licensed or vaccinated, the owner might have to pay for that too (depends on the shelter, and the local laws). It might cause the owner enough inconvenience to make them keep their dogs penned up, or they might decide it's not worth paying to get the dogs back.
(Of course this does not work if your area has no animal shelter, or if you cannot safely deal with the dogs, and it would probably take more of your time than shooting the dogs.)