I can't fly at night, but at Beedle's Island, he says that he does business there at night. I don't see any beds on the island as well. How can I get there?
There is a bed on Beedle's shop that you can sleep on. Once you sleep there until night, Beedle will park on the island and you can then do business with him (and also find a Gratitude Crystal there as well.) Sleep in the bed again so you can return.