On my first playthrough I chopped off the leg and sided with Kenny in the meat locker.
I cut the leg off immediately since I was certain it was the only way. I don't really see why we weren't given the option to cut down the tree though; then Lee, Kenny and Mark could just have carried the teacher, the trap and the chain into the camp. I'm sure they'd have found use for the chain in the fortification process as well.
As to the meatlocker, I don't know how other people manage to bring up any positive feelings towards Lilly whatsoever. To me she's just completely unlikable and whiney, and the few times she's "showing her soft side" or being nice just seem utterly artificial, which caused me to discard them from my general view on her entirely. Occasionally I found her even worse than her dad.
So yeah, I helped Kenny because I'm willing to do whatever it takes to piss Lilly off and, hopefully, speed up the ETA of her leaving the group. Apart from this I've so far always preferred Kenny's views on things, and in the meat locker he was making a lot more sense than Lilly as well.