At least you're not in suburbia, so you actually can shoot pests. I keep running wabbits out of my yard, and they keep coming back.
What about a decent air rifle?
A fair chance of neighbors seeing me & raising hell about it. But I am liking the havaheart then slow cooker idea...need to check the legalities thereof...
Or just put up a high perch, inviting the caracaras to hang out...