Where to stay in Mt. Fuji area for 1 night

In August we will be travelling to the Mt. Fuji area from Tokyo by train and spending 1 night in the area and am looking at renting a car once in the area to drive around to the various areas. I am trying to determine if it would be better to stay in Hankone or in the Kawaguchi area on the lake front and where would be the most convenient place (i.e. Fujisan train station) to pick up a rental car. Which of these areas would have more options for doing things at night such as restaurants / bars ? We would like to cover as much of the area as possible and see the best views of the mountain from the viewing platform and lake front, ride the ropeway, do the lake cruise and see some of the museums in Hakone. Would it be best to try to arrive in Kawaguchi, pick up a car, drive around on the first day and then on the second day go over to Hakone area and return to Tokyo by train from there later in the day. If so, does anyone know of a rental car agency and location where it would be convenient to pick up a rental car in the Kawaguchi area and return it to the Hakone area where it would be close to the train stations. Thanks for the help.

If the return place of the rental car is different, extra "drop off fee" will be charged. (as oneway use)
If possible, you can rent at Kawaguchiko station and drive
to Hakone Yumoto or even Odawara, which is a bit closer to Tokyo.

If you want to rent a car, you can stay far fr the rail stn without problems.
The view of Mt. Fuji is also good on the side of the Five lakes.

In Hakone, you can rent rare sports cars that appear in famous Japanese comic/anime.
I don't write the name of the rent shop, search if you are interested, they have EN page.
However, you can not "drop off" at this rent shop.
If you or your friends like cars well, should be proud this rare experience.

(small note)
Pls don't use/call "Kawaguchiko area" as "Kawaguchi area", bc the name of the town is not also "Kawaguchi",
https://en.wikipedia.org/wiki/Fujikawaguchiko,_Yamanashi
and same name famous city "Kawaguchi" located in different northern-west part of Tokyo area.
Most Japanese doesn't get angry with that, but it is confusing and not right as town history.
https://en.wikipedia.org/wiki/Kawaguchi,_Saitama
Last "-ko" means just "lake" in this case.
https://en.wikipedia.org/wiki/Fuji_Five_Lakes