Gotta put her down. In this world, not safe to keep her around, and no time/resources/safety to deal with her mental situation to 'cure' her.
Do you think Tyrese eventually came to that more logical conclusion at the end when Carol admitted to killing Karen in light of the fact he also witnessed her killing the girl?
On this episode - this was IMO that best acted and best story arc in the whole series. Hats off to both Chad Coleman who plays Tyrese and Melissa McBride who plays Carol. They are as good if not better than Andy Lincoln in some respects.
One of the amazing things I think this episode covers - very well in fact - is not many TV shows will touch mental illness and especially with children. In fact the only other time I can think of when they've touched mental illness in children on TV is a Law and Order episode where a young girl was trying to revive a boy she had killed with a battery.
Obviously Lizzy had some sort if dissociative disorder - one in which on her plane of existence the walkers are more, "real," than humans.