(Not a lawyer, opinion here) So is she AirBnBing the whole place, or is she living there and just doing a room? That would make a difference in how you handle this. In any case I would do the following:
Screenshot her AirBnB listing of your place just so you have some records that is going on. Most leases include a prohibition on running a business out of a rental, but I assume you've already looked at the old lease. You'll also want to explicitly cover that in your lease template, so you'll need to work with your lawyer.
Read the eviction moratorium and contact a lawyer. The December renewal actually isn't that restrictive and if the tenant didn't come with a Declaration that they are a covered person (which would have come from the prior owner) then they may not be protected. Sounds like this is a situation where they just don't want to pay, not that they can't because of covid. Just being stubborn isn't covered by the moratorium. The lawyer would be able to tell you what the local steps are (including steps short of eviction in order to encourage a new lease signing).
Don't accept rent for January for the old amount before talking to your lawyer! You might inadvertently create a month-to-month tenancy under the old terms.
Good luck - let us know how it goes!