You're correct with your assumption that I found lodging on the way. The sign pictured in the second photo says Minshuku Kamagari, and its the name of the place I stayed along Tobishima Kaido on the way to Hiroshima. I also stayed a night at Guest House Tokonoma, which was great. I also stayed in several bike friendly business hotels that either had an indoor bike parking space or let you take your bike into your room. I think you'll find a great range of minshuku, ryokan, and hotels to choose from throughout Kyushu, and most will be bike friendly.