When I was a kid, we had trouble with dog packs hassling the sheep. A few were killed--the sheep--and had to get neighbor out to help with the dog attack once and one was shot. Damn we felt bad for that but the dogs were running wild and we'd tried everything else first.
Beating it to death it just plain evil. And I'd tend to believe the eye witness. You don't dream up a trap on the dog's leg easily, especially when going against the powers-that-be in your area.
I hope someone somewhere investigates properly. Was the trap even adjacent to the coop filled with chickens? Because that's the only way that action could make sense--that you set traps immediately outside the chicken coop to try to trap what was killing them, especially if you didn't know if it was coon or fox. But just because a dog is in the trap doesn't mean it did the killing either. It might have been following the trail of the actual killer.
But it sounds like you also have witnesses saying THERE WERE NO CHICKENS. So how do you kill an animal killing your animals when you don't have any?
Anyone have a good telephoto lens and can get pictures of his coop without chickens?
Also, spread the word in your local area to take pictures before calling 911 if discovering animal abuse. Take photos and upload to cloud then call 911. It sucks for the animal to wait a moment longer, but then at least you have timestamped proof of actual abuse.