I really don't know about the best way to cull (probably the head removal if you can handle it) just wanted to tell you a story so you feel like yes, you are doing the right thing. My mother has a handicapped chicken. When she came in an order we split she just seemed weak but quickly progressed to what you are describing. I asked her over and over to let me put her down and she never did. She is now 9 weeks old and has never taken a single step. While I wouldn't dream of culling her now, she is a sweetie and now we are all very attached, I know it would have been better for everyone involved if she had done it when she was tiny. Right now I am trying to figure out how to make a permanent wheelchair type device for this chicken so she doesn't have to always lay down. Good luck and I am sorry you have to do this. Just know you are doing the right thing.