Sorry you may have to do this. I'm kind of the family "putter downer" for whatever animal needs put down or for strays that they don't want wandering around. Its not especially fun.
In this case, chopping the head off with a hatchet would do pretty much the same thing as wringing the neck, and I would be more comfortable doing that. But depending on how you do it (you're fairly limited) and how nervous you are, you have a chance at chopping your fingers off. Be careful.
I don't know about the humane factor, but anytime I've killed a chicken it's been with a .22 to the head. You have to be a good shot though, and make sure your rifle is sighted for the range you are shooting at. One benefit to this is that you don't have to catch the chicken or be in contact with it as it dies.
Whatever you do, the chicken will appear to be alive and flop around for a bit even after it's dead. Be prepared mentally prepared for that if you are attached to the chicken at all, as a pet.
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