The part of Shinjuku to Kawaguchiko is not interesting enough. The direct bus on this route could be enough.
From Kawaguchiko to Yudanaka, there are Lake Suwa, Matsumoto (castle), Nagano (Zenkoji), and Obuse (village). All of them are doable by trains. I mean just getting back to Shinjuku would be nothing. In summer, I prefer to go over Mt Yatsugatake, skipping Lake Suwa and Matsumoto.
From Yudanaka to Takaragawa, you go over Mt Shirane and Kusatsu. That would be more appealing than the Yatsugatake route.
You could come back to Shinjuku on the Kan-etsu Highway from Takaragawa. But, it would be just boring. I would suggest making a detour to Nikko. There is another Mt Shirane in the Oze and Nikko area. Okunikko, having Lake Chuzenji, could be more appealing than the Kusatsu Shiranesan area.
From Nikko to Shinjuku could be boring, too. But driving in Tokyo highways in the night time could be a fun.
So, my preference is
1. Nagano - Yudanaka - Kusatsu-Shiranesan - Takaragawa - Nikko-Shiranesan - Nikko
2. Kawaguchiko - Yutsugatake - Nagano
3. Tokyo highways
4. Shinjuku - Kawaguchiko, Nikko - Shinjuku
I might limit a driving from Nagano to Nikko, or from Kawaguchiko to Nikko. Both Kawaguchiko - Shinjuku, and Nikko - Shinjuku cost less than 2k yen by public transport. But it would be easier to handle the luggage by doing them all by a car.
The driver's seat is on the right. Cars run on the left. Are you getting to used to them? Roads are rather narrow in a mountainous nation. In August, the traffic could be busier in the country sides than in the cities. Especially, in mid-August, there is the Bon on Aug 15 and the mountain day on Aug 11.
http://www.japan-guide.com/e/e2286.htmlhttps://ja.wikipedia.org/wiki/%E5%B1%B1%E3%81%AE%E6%97%A5
