Mon Travel from shin-Osaka to Shin-Yamaguchi take bus to Akiyoshidai spend night in Okayama
Tues go to Naoshima and then spent night in Fukuoka
Wed unsure lots of options, possibly BioPark/Huis ten Bosch? See some of Fukuoka?
Thurs go to Kurokawa for the night
Friday drive to Takachiho Gorge sightsee and return to Kurokawa for the night
Sat Drive to Kumamato Airport and fly to HND with Airpass.This option 1 seems to be very rushed. And I donLt understand Monday. Do you go to Shin-Yamaguchi in Yamaguchi prefecture and then spend the night in Okayama which is way back towards Osaka? Just to go on the next day to Naoshima?? Sounds to me like a bit of a crazy trip.
Also Naoshima and then night at Fukuoka in general seems to be quite challaging. I believe you said on a previous post that your daughter is not a night person, so I guess you will want to arrive at Fukuoka relatively early, but from Naoshima you first need to take a ferry. Naoshima is a very nice place and you could easily spend there two days (one night). There is even a small beach. Nothing that an adult would be totally enthusiastic about, but I could see that a child could enjoy it quite a lot. Its close to the Benesse house.
So maybe Monday to Osaka to Naoshima, enjoy the island, sleep there, then enjoy more art on Tuesday and Tuesday after lunch leave for Honshu and on to Fukuoka.
or
Mon visit Himeji, maybe Korakuen (I've been) and then go to Naoshima see Benesse House and the pumpkins which might be quieter since nothing else is open and overnight
Tues see everything else in Naoshima early and then head to Fukuoka for the night
Wed From Fukuoka take the shinkansen 30 minutes to Shin Yamaguchi, see the caves and then go back to Fukuoka,
Thurs unsure of what we'll do earlier but take the shinkansen eventually to Kumamato rent a car and go to Kurokawa for the night.
Fri drive to Takachiho and go back to Kurokawa for the night
Sat spend morning in Kurokawa and then fly out from Kumamato AirportThe Monday would be still to busy for me. Your have Himeji, Korakuen, Ferry, Benesse House and pumpkins. That`s quiet a lot for one day. Himeji is nice (at least it was nice 20 years ago when I was there). If you go to Okayama, my personal favorite would be to stay 2 or 3 days there and see Korakuen, then do some light cycling around Kibi plain (I remember your daughter is not an experienced cyclist yet), but in Kibi plain, it is really a plain and the streets are very rural, no cars, so very safe for kids as well, and you get to see some temples and shrines. And a half day in Kuroshiki maybe combined with Takahashi castle, which is a very nice castle on a hill top.
https://bicycletraveljapan.blogspot.com/2017/11/day-46-okayama-and-tak...https://bicycletraveljapan.blogspot.com/2017/11/day-45-kibi-plain-and-...
or
Mon go see the caves and overnight in Fukuoka
Tues-Sat do Kyushu things like Kurokawa and Takachiho before flying out from either Kumamoto or Fukuoka airport.This seems to be a much more relaxed plan. You could visit Dazaifu, Nagasaki, maybe go down to Kagoshima, just enjoy 2 days of relax in Kurokawa, get close up to Mt Aso (if it is open then), and you will need a car there...
I don`t know if your daughter or you know Asterix, but if you do, a visit to Yoshinogara in Saga prefecture could be nice:
https://bicycletraveljapan.blogspot.com/2017/11/day-53-yanagawa-to-sag...https://en.wikipedia.org/wiki/AsterixThere is a curious museum for toilets in Northern Kyushu. I havenLt been there, but it definitely sounds interesting.
Enjoy your trip to Japan!