*Edit: Sorry, my mistake. This is for Seoul, rather than Jeju. I will leave it here anyway.

Mr. Yi Dok Yong at Bikely can help you out. He speaks good English and is responsive to contact.

Email him at: bikely@bicyclestation.co.kr

I think he has some of his own cromolly freerange bikes (?? ???? : Free Range - Free Range ???? ??) available for hire. They come with racks and Ortleib panniers AFAIK, but do not take my word for it, as things may have changed or they may be booked out already.