I think it is a great thing to do. I know a guy who took his family on two BMW motorbikes (husband and wife each on one bike with kids on the back) to Kyushu.

Tolls will run nearly $20 per 100 km. My guess from experience.

Shinkansen is not cheap either especially with many people. You get a skewed and wrong impression of Japan if you travel by train.

One problem is car rentals in Japan are very inflexible (well expensive) re one way rentals. Instead what you could do is drive around Kyushu and/or Hokkaido.

If you have airline miles then it's very cheap and easy to fly point to point and then get a rental car for the local area. You probably don't need one around Tokyo.

p.s. Finding parking in Japan is very easy. It can get quite expensive in a place like Tokyo. In the country hotels will have free parking. It's not that congested outside metro Tokyo and Osaka.