Getting from Hakone to Lake Kawaguchiko
 

We will be in Hakone for 2 nights. I really wanted to spend a day going around the scenic spots near Lake Kawaguchiko. However, all the get your guides and Viator groups seem to start from Tokyo. Does anyone know a private driver we can use to drive us to the lake area from our hotel in Hakone?
Alternatively, even though I really don’t want to have to drive “on the wrong side of the road” any tips on renting a car for a day? I saw Toyota Rent a Car and Nissan Rent a car are nearby. The Toyota website though says they are pretty much closed for the next few months which is strange!
I am also open to using a taxi but don’t want to get stuck. Also, cost?