i understand where you are coming from and like you I would want to clear my name, but you may "rock the boat" in doing so.
I would leave it to setlle for a while and ask someone in the police force how they would deal with this. get you feet firmly in and make some good friends who will be able t help as apposed to making things worse.
You rneighbour has unfortunately already won the first battle, by makling you angry. Dont give them the satisfaction. Call their bluff. If you are innocent, which I think you are, then you have nothing to worry about. If you didnt trust the judicial system then you wouldnt have joined the police. Call their bluff and laugh it off. See their face when you turn up home with your police outfit on. Im sure she will have second thoughts on making up a story and calling 2 witnessess then!