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We had a raccoon problem this summer. We live-trapped FIVE enormous raccoons in about 2 weeks. To dispatch them, I used a quiet, discreet method. I covered the trap with a large piece of construction grade plastic (a tarp would work fine too) and then covered that with an old bed comforter. I weighted down the sides of the comforter with rocks to keep the cage fairly air-tight. I then backed my car up to the vicinity of the cage. I attached a piece of drainage piping to the tailpipe of my car and secured it to the tailpipe with aluminum foil (works great). Then I tucked the free end of the drainage pip under the plastic/comforter and secured it on two sides with more rocks to keep the exhaust inside the trap. I started the car and in about 15-20 minutes, the raccoons were asleep. Permanently. It's the only way I could've done it. Other methods just wouldn't work for me for a number of reasons.
Sorry about your chickens and good luck with the predators. We all learn hard lessons about predators as chicken-keepers, but the key is to learn the lesson from the breach in security and make sure it cannot happen again (or it will).
Good luck.