I'm so sorry to hear of the loss of your egg laying bird! Is it possible to knock out the bird first, maybe with a mallet? I never thought about this part of the whole chicken raising thing. Just figured they'd die out naturally. Miltex makes some really nice scissors. During the hatch I used a small pair of Miltex Iris scissors to puncture the shell and remove small segments, however the chick had already expired. Iris's may be too small for this purpose, usually they're used to delicately dissect soft tissue. Perhaps a veterinary person would know of a better instrument for this purpose? Is the technique to sever the vertebral body mid level of the neck?
There have been a few ways to put a chicken down but alot believe a clean cut through neck by head is best. I've done a few that people say are humane and decided as much as I don't like this way it works. One other way is vinegar and baking soda. It sorta knockes them out by replacing oxygen with co2. When I did it, it seemed like it went to sleep with one death flop and it was done. This was a chick about 1-2 months old I think. I have also taken a shovel and hit a 3ish month old chicken over the head it wasn't moving around much had trouble standing and it was this winter and what I had ready to use. It didn't flop around ether but it was already weak. It seemed fastest to me but if you don't hit hard enough or in right spot you could just injure it and make it run away. So I've gone with the removal of head. It seems the easiest and most accurate and humane to me but make sure you have sharp tools that can actually cut through the neck.
A few that I've not tried and don't plan on it. Is putting chicks in bag then in freezer. a box and a tailpipe. a sealed bucket. I think some do a cone method where put chicken upside down I a cone and just slice/puncture the juggler and let them bleed out. But I think that's more if your going to eat the chicken. And the only reason I don't do that is I don't have a cone and I don't know where the juggler is so removal is best for me.