We did 6 day tour in Kyushu over last thanksgiving. Two words of advice - do it! One of the best trips we've done and we travel a lot. I am sure we would be cycling for several days next time we go.
1) We chose Kyushu because it was November and it's the best weather there that time of year and we wanted to visit a few onsens and Kyushu is the best place. For june, it is perhaps not the best choice. But if interested, can share our itinerary.
2) Bikes: local rental is hard. I've emailed perhaps a dozen companies locally and only found one that was willing to rent what they call 'sports" bikes, mountain bikes with disk brakes. No racks no option to change pedals etc. From this perspective
https://globalwheels-japan.com was a better choice. they provide all the accessories, can provide rinko bikes for trains, racks panniers if you want etc. It was about 2x in price (including shipping charges) vs the local place but I think it was worth it. However, [we most often ride tandem at home which are impossible to rent] and we ended up purchasing a used Bike Friday tandem and rode it. It almost paid for itself in this one trip as compared to the rent cost.
That being said, there are multiple bike rental place around Shimanami Kaido. I would suggest doing it in 2 days with loops around each islands - there are plenty to see. This was our original plan but we thought it could be too cold and windy in November. You can then continue along Shikoku cost and then figure out a ferry/train return option to give the bikes back. I think Hokkaido also has some bike rental options but could be a bit cold in June.
3) Bikes on the train: it's a hassle but if you get a gist of it it is not that much of a hassle. I think getting a motorcycle cover and duct tape from 100Y store will get you better packing then a standard rink bag. So don't be afraid of open jar itinerary or making some transfers on the train.