I know chicken ranchers who use #2 treble hooks to deal with coyotes. Tie them to a nice, sturdy tree branch with heavy deep sea fishing line, leaving the hook hanging about five feet off the ground. Wrap a big ol’ piece of raw chicken meat around the hook, making sure it finds purchase on all three points. Then check back every day or two to see what you’ve caught.
It doesn’t require any fancy shooting hardware, nor even human intervention. Just put out the bait and let nature take its course.
Heck with your set up, you don’t need a firearm, just a really sharp knife and tough stomach.