Perhaps there is a way you could just release it somewhere far from your coop like the woods or a park? It’s already in a live trap, why kill it now?
I'm sorry, do not every release a captured vermin so that it become somebody else's problem.
I have enough fox, raccoon, rats, mice, voles, bob cats and cougars around us to last a lifetime.
I would be really POed if somebody dumped their nuisance pest outside our farm gate, and in the same breathe, I doubt if you would be overjoyed if somebody did the same to you.