When I was a kid, living on a turkey farm, we trained our Weimaraners to tree the many unwelcome feral cats that people were always dumping on our property. (Cats are extremely hazardous to an intensive poultry operation, for many reasons). There was often a danger of putting a .22 bullet both through the cat and through the roof of the barn. We solved that problem by creating very large hollow points with the tip of a pen and then cutting a shallow x in the tip. Accuracy was never tested, but at 20 feet they were very deadly and did not exit the cat. Today I'd probably use CCI Stingers with a bigger hollow point for that job.
I caught chickens once when I was young. Notice I said once.