When I was in Jinmen, the military headquarters ended up being the landmark that I kept coming back to. Not only were there free bathrooms inside of it, but it was located somewhat in the middle of the town closest to my guest house. The military quarters came into use during the Qing Dynasty and were used for various purposes over the years (more on that in the Wikipedia article). Now they house a small museum, complete with wax figures. These are lanterns that hang in all the hallways. As you can see, the headquarters are open into the evening, making the bathroom component especially convenient.