Neighbors will get a cat, and let it roam free, which I don't like since I don't want the cat to kill birds. We will see the cat for a few months and then the cat is never seen again. A few months later, a new cat is seen, but it too disappears after a few months. This repeats over and over. With the foxes, coyotes, owls and hawks, the cats don't last long.
I saw a video years ago where a guy had a company for killing rats and other vermin. If I remember right, he was using trained minks or ferrets, to go into the boroughs and kill the vermin. The animal was killing vermin had an incredible rate and would have to cool down in a water bath after a period of time.
Maybe there is someone with a similar company in the OP's area.
I had a dog that was part terrier and it would kill birds if he could catch them in bushes. He did not have access to rodents but I think he would have killed them if they were around. That dog was very territorial and had no fear.