Sounds like a good plan, i feel a little jealous as when i was in Fuji i arrived just before the season to hike

In that regards i can't offer much advice since i didn't actually get to do it, however what i did do and recommend you also do was renting bikes and cycling around the countryside along the lakes. Me and my friends cycled a good amount of kilometers and spent the entire day doing it just cycling, stopping and enjoying the scenery. The surrounding area is absolutely fantastic and beautiful place.

I also recommend staying in K's House Fuji, it's where we stayed for the duration. Excellent accomodation, friendly staff, lots of room with free pickup from the bus/train station. We rented our bikes from them as well

P.S There is a Public Bath/Hot Spring establishment just beside K's House, it was our first time on the trip using one!!