My initial reaction was to suggest you drop the car off at Osaka and use the train for the Kobe-Okayama-Hiroshima-Fukuoka leg.
But although the Osaka-Hiroshima leg would basically be driving from city to city on the motorway, with little to see in between, making the train really much faster and more convenient, a really fantastic route (time permitting) would be to put the car on the ferry at Hiroshima to Matsuyama in Shikoku (allowing you to check out the castle and Dogo onsen), drive to the small port town of Misaki in the far west of Shikoku and again ferry over to Oita, from where you can do the stunning drive across to Kumamoto castle, taking places such as Usuki, Mt Aso, Yufuin, etc along the way, and then grab the motorway up to Fukuoka...
I would recommend you use the train for your day trip to Nikko - it takes a couple of hours on the bullet train and will take you at least 3 hours by car, even using the motorway (that's six hours driving there and back).
For the rest, I would do it in this order:
Tokyo-Hakone-Kawaguchiko-Matsumoto-Takayama-Shirakawago-Kanazawa-Hikone-Nara-Koya-Osaka/Kobe.
(Btw, if you are lucky with the weather and getting good views, there are some seriously gorgeous drives around the Hakone-Kawaguchiko area, such as the Izu Skyline, the Ashinoko Skyline and Mazda Skyline. Similarly, when in Shirakawago, be sure to check out the other villages such as Gokayama - this is when having a car really comes into it own, and you should take full advantage of it...).
In 2015 and 2016, I did a 46,000 km driving trip round Japan in my campervan. You might find some useful info in my blog (I'm afraid you'll have to scroll through to find the relevant parts of Japan for you)
https://myjapantravels.wordpress.com/However, be warned that driving in Japan can be painfully slow once you get off the motorway. Do NOT take google maps driving time estimates as being anywhere near accurate!! You need to add on at least another 50% to whatever time they give...as an example this weekend I did a day trip to a place 70km away. According to google maps, it should take about 2 hours. In fact, it took well over 3 hours on both legs there and back...