Try contacting your local Wildlife department and see if they can help. Other than that you might try to bait the cat then shoot it when it comes too eat. I don't know if this works with cats or if it is legal in your area but it has been known to work in our area on problem coyotes.
Best of Luck!!
Ditto on checking with the wildlife department. They should have good advice on how to accomplish the feat. More importantly, if you do it the wrong way you could find yourself seriously afoul of the law.
First make sure your gun is safe, remove the bolt and clean your barrel from the chamber to the end of muzzle. Clean the barrel using butchers bore shine until a patch comes out white. Run a total of 5 separate patches soaked with denatured alcohol through the bore and let dry for 30 minutes. Insert a cleaning rod with a jag and a dry patch into the neck of the chamber. Make sure it will stay there through the next process. Now using another cleaning rod, install a patch that has been coated...