I had at least one playthrough where she decided to shoot Lee on her own. I called it my hardass playthrough. I killed just about everybody I could except the Stranger (Wanted Clem to do it), the woman in the street (Better for our survival if I didn't), Fivel (Wanted Kenny to man up already) and Danny St. John (Because It was fun to piss Lilly off after killing her dad.
I constantly told her the truth about everything, told her killing the St. Johns were necessary, her parents were dead, Lee killing before didn't really matter anymore, brought her to Crawford, never downplayed dangerous situations. I also let her vote on Ben, the immediately voted to throw Ben out afterwards. (Her face was priceless.)
In the end, she decided she could do it. Not sure if you have to do all that, it's just a few triggers or just random, but it is possible for Clementine to decide on her own to shoot Lee. Maybe it's not so much about what you teach her but how much of an asshole you are to her. I'd imagine that would make it easier to shoot Lee.