Trying to tackle Fukuoka, Oita, Miyazaki, Kagoshima, Kumamoto and back home via Oita is a huge distance to cover in 4 days on a big bike fitted for long distance touring at 400+ KM a day. You'll have to significantly reduce that while traveling by scooter. Something realistic (and something I would highly recommend) would be spending a couple of days in Kunisaki. It's going to take you at least an entire day just to get to Kunisaki, and an entire day just to get home from Kunisaki (especially since it's a 3 day weekend) and there's enough in Kunisaki to keep you mighty happy for 2 days. There are lots of campgrounds in Kunisaki too.
It's 43 km from your house to the campground in Akiyoshidai. It's 61 km from your house to Hagi, and 64 km from your house to the bridge at Tsunoshima.
It's 119 km (twice as far) just to get to Usa Jingu which isn't even the Kunisaki peninsula yet. I suspect that even getting to Aso and back home in the amount of time you have would be a superhuman effort.
Here's a campsite where you can stay at a tent site for 540 yen a night. Just pitch your tent and leave it there, ride around for a day enjoying the sites, and return to the same site in the evening.
If you wake up before the sun, and ride all day stopping only for fuel and food, you might be able to make that campsite in one day. Same with the return ride.