This is such a stupid problem to have....
Completely not your fault but because of someone else's irresponsible reckless actions you are loosing your chickens. Not to mention your own cat was attacked.
There is no such thing as too far out in the country for stray cats, I call them drop offs. Everybody deals with them differently, my closest neighbor live traps them and takes them to the vet to get fixed. His wife likes cats so he deals with great expense doing this. We have a lady about a mile away who keeps strays in her barn, every few years someone calls animal control and they cart 50+ cats away. I use the lead approach, let's be clear these are not the " neighbors" cat that happened into our yard. This is farm county not suburbia, I think that gets lost on this forum a bit to much. Most don't survive my dog honestly. We saved a kitten last year I found living in the garage in February, she couldn't have been 3 months old. The meanest nastiest little thing you've ever seen. My inlaws took her in, to this day they have to trap her to handle her, it's ridiculous.
We have tried other approaches but the only one that works is zero tolerance. Hopefully you will get this problem solved soon, and not have to deal with again. Next time you see a strange cat you know what to do, it's not worth the risk to your animals.