You are on the right path in your thinking. Protect yourself by using the police to get your side of the dispute on record. Show the neighbor his empty accusations will be met with diligence on your part to protect your rights. If the police continue to record that you are not the person involved in the bb throwing and his are empty threats, soon they will be important allies for you.
Next steps beyond this will require more details so perhaps see a local lawyer when the time arrives to do so.
God bless.Ask a similar question