Awwwww poor little twerp. 
Beautiful chick. How does it seem to be doing?
Culling (putting them down)...there's different thoughts, different ways. IMO the BEST way is the quickest and most painless. I've only culled if they seem to be in pain. I've had a few that nature took quickly, but they seemed comfortable (I had a muscovy duck I assisted halfway with hatch....very deformed bill, I let it sit in it's shell only halfway hatched to give him a chance and he passed peacefully). I had a turkey poult who had intestines fallign otu and was in obvious pain. I tried to "gas him" in a plastic bag using the exhaust of the car, which I had heard was quick and painless. It was horrible. I stopped and resorted to a shovel. I sobbed. But, he was so tiny and it was fast for him. The shovel part. Not the gas. One of his hatchmates, now getting very big, I found near death about a week ago. I brought her in the house, tried to warm her, tried to give her a few drops of sugar water but it was obvious she was dying. I put her in a cardboard box wrapped in warmed towels and let he rpass peacefully. I only hope she was comforted by the warmth. We recently had to get rid of one of our favorite roosters (he turned really mean), but I couldn't do it...so I gave him to a young couple who were going to eat him. I'm only comforted knowing he lived a good life with us, and would go on to nurture another family. So, there's all sorts of reasons for culling. When they are newborns, I think it's "easier" in a sense. You know it's usually because they are sick or deformed and not having quality to their life. Some people use sharp shears (lop off their head), and will cover with a towel while they do it. A shovel to the head will crush their brainstem, also quick and relatively painless.
I hope your little chick can make it...as long as he does not seem in pain it's worth a shot.
