Fr Chuck I think it might depend on where you live but you are right.
Our 4H Rabbit leader's rabbits were attacked by dogs. They had to pay to replace each rabbit and the loss of future income from that rabbit. Since they were 4H leaders they had records to prove how long a rabbit lives, how much they could get from each batch of babies, and how much the rabbit was worth. They were also able to show how much a show rabbit brought in at auction. Whenever their neighbor's dogs attacked, they ended up in small claims court and they walked away with $$$ all the time. I seem to remember them proving one rabbit, that had won Grand Champion and they were using for breeding brought around $600.
Once they had proof that the court was on their side, neighbors with rabbit-loving dogs were more willing to keep the dogs under control and if there were problems, the family had no problem just negotiating a price to avoid court.
I would assume that you could do the same with chickens.
The area also had signs posted all over that Dogs found molesting animals were allowed to be shot. This was in CA.