What I see lacking here is any kind of proof, other than the poster's word, that the ONLY possible source of damage is the exit from the parking lot. How does the school know that you didn't hit something else, somewhere else, were not seen and thus never reported it? How does the court know that? How do we know that, for that matter? Yes, I know that's what you SAID, but you have a vested interest in having someone else be responsible. (No, this is not an accusation - I'm trying to make a point.)
Do you see why it would be a difficult win?