As an educator, you should already know that people file false abuse reports every day, and that the authorities are required by law to investigate every single complaint, even if they are convinced it is frivolous.
If you're concerned about the report showing up on a background check, have a background check run on yourself now - there are numerous services available online - and see what it says. It's unlikely that unfounded allegations of abuse are going to show up, however.
At the moment, you really have no provable damages. It's hardly worth the effort of trying to press harassment charges if she's moving next month anyway.

