I'd also question what the point is for spending 2 days in Osaka, when you say you've been there already. There are some great places at night like Dotonobori, and the night views from the Umeda Sky Bldg and Abeno Harukas Bldg are mind blowing (and not the same).
https://www.youtube.com/playlist?list=PL7LcPcmoSg-jkgQdiQlK2bcKCzr2pJQ...Spending a night at Koyasan would also be very worthwhile.
It's also amazing that nobody above has mentioned seeing Himeji, which has the finest castle in the whole country. You could make a whole day trip seeing the castle, Kokoen Garden next to it, and Mt. Shosha. You could use a Kansai Area Pass for a day and use a 7 Day Pass for the longer travels.
Okayama is not far and it is famous for its castle, plus Korakuen
https://www.youtube.com/watch?v=5zAI3_N7NEM Garden which is one of Japan's Top 3, and also the Handayama Garden is beautiful:
https://www.youtube.com/watch?v=JGQ8o7k9GbA The half day hike up to Konpirasan in Kotohira, Kagawa
is magnificent.
https://www.youtube.com/watch?v=ley9rVIewoQGoing to Takamatsu is also nice - it has one of Japan's finest gardens also, called Ritsurin.
https://www.youtube.com/watch?v=afx-Ju52FiYHiroshima also has a number of good places with Miyajima being the most famous. Everyone goes to see the Itsukushima Shrine with the 'floating torii', but it really doesn't take long and the real gem is the Daishoin Temple - a place you will find unforgettable.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-iCNZ6AdsKGUB-HPKNxP1...And aside from Hiroshima City itself, Onomichi with its temple tour and Kosanji Temple is spectacular.
https://www.youtube.com/watch?v=97Q5EjEOXI8Just beyond Miyajima also lies Iwakuni with its historical bridge and mountaintop castle.
https://www.youtube.com/watch?v=0COOZXi66mMAnd if you really want something mind blowing, go see the Akiyoshido Limestone Cave in Yamaguchi - a 300 million year old marvel with an underground river going through.
https://www.youtube.com/watch?v=DqIKC9S8TFQIf you want to base yourself out of Fukuoka and do several day trips to avoid lugging stuff around, that is fine. There is a lot to see in the local area, more than most realize. To see what Fukuoka has, you could have a look at
http://www.thejapanfaq.com/fukpic.htmlhttps://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...You could also see Kitakyushu a bit along the way down - Kokura Castle is not bad, the Moji Retro area preserves a Japan from a century ago that's now long gone, and the view from atop Mt. Sarakurayama is a secret jewel that none of the tourist guidebook "experts" even have a clue about.
https://www.youtube.com/watch?v=Hl1pCW9IbbkYou could do Nagasaki as a day trip, but better to do an overnight there and spend the time to see all that the city offers.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-gbw2yXNLK92siHyNqSNx... is stunning and takes up 3 hours alone, but it's like walking by Chernobyl without the cancer.
Kagoshima is also easily reached by the bullet train, and Sakurajima is a place you'll be quite amazed at.
https://www.youtube.com/playlist?list=PL7LcPcmoSg-irtXXB8IJMdkB9mF0qjn...Going to Kumamoto and Beppu are also worth considering. Kurokawa is also superb but you'll have to pay extra for a bus or renting a car.