My dogs, some, have demonstrated a capacity to kill neighbor's stock then return to defend ours of same species. Your dog could kill even a large turkey or goose in short order even though generally timid. One of my wimpiest dogs had no trouble engaging even large healthy boar coons and winning. That being said, some wild animals, especially the foxes, will surplus kill (would appear to be for fun until carcasses policed up by predator).
Neighbor nearby targeted recently by my dogs was paid for losses. I also pointed out their stock left their property and when that was contested was able to walk them around house to point out flock walking down road in real time. Legally I was responsible only for losses where my dog came onto their land and most were not on their land. Those on their land I was legally responsible for. As conversation with that party continued it became apparent they had been loosing a lot more birds one at a time for an extended period of time (wildlife caused). I recommended they adopt a more sound fencing option to the deer netting in use and suggested they use electrified poultry netting or hotwire; I use both in concert with my dogs that know how to navigate it keeping losses of free-range birds very low. At some point like with other poultry keepers in the area, a predator they do not currently know (likely to be a dog) is going to depopulate their flock. The bone-headed attitude with insistence that only the dog owner is responsible increases odds such poultry keepers either drop out or spend a lot more on replacement birds until improvements by poultry keeper are made. Getting neighbor to adopt more effective and costly protections is delicate, especially in the aftermath of an attack, very especially if they attribute losses to your dog. You will need a thick skin when dealing with such and go to great efforts not to be offensive. This something I have a fair amount of experience in as not writing hypothetically or as what should be done.
That being said, OP, you need to keep dog on your property when there is even a suspicion your dog is involved.