No, it doesn't. I was re-reading the Growing Hunger rulebook last night, and I believe it was in the FAQs in there that it clarifies - when the Fence Post is used, the fight technically "takes place" in the Zombie's square.
So in the situation you describe, Lizzie does not affect the results. But if the roles were reversed and the Zombie was inside the building and the Hero outside, I think that would work.
oh shoot, you're right. I didn't read his entire post. It technically would be a Fight, but it would take place in the Zombies' space as you said.