One of our cats kept getting out of our old house - took us a couple of weeks to work out that she'd found an access hatch into the subfloor that we didn't know about in the back of a cupboard and worked out how to lift it up
If your cat is attacking neighbor animals you should not let it outside anymore. Culling wildlife is problematic but "their nature". But a pet that actively attacks other people's pets is just a menace and will only end in heartache and legal issues.
Maybe make the cat indoor-only and save on vet bills. And save the animals in your neighborhood.
It's a fact that cats hunt for fun in addition to sustenance. Since you presumably feed your cat, we can only assume that everything he attacks outdoors is for fun.
So if I run your cat over in the street, you would have no problem with it? Because you sure seem okay with letting your captive predator out to do the same to your entire neighborhood.
Cats being predators are meant to kill things. If this causes a problem in your area, it's a sign cats weren't meant to be in that area and you should probably stop buying them and domesticate some native wildlife instead.
Our cat brought a live bat into the house once and then just let it go. The dogs were all over it, but thankfully didn't kill or injure it. I let it crawl along the wall to the front door, then it climbed up the screen door. I opened the screen and closed the front door. We checked back a half hour later and it was gone, so I guess it was okay, and flew away. That was a crazy experience.