Crtain things you have, you keep when you play the song of time (ex. all of your masks, the equipment you've gained, and the remains of bosses you've defeated. Plus you still know the songs you've learned.)
The game only saves in 2 ways. 1, it saves whenever you play the song of time and go back to day 1. if you play the song of time, do stuff, then quit, itl just bring you back to when you first wen back to day 1. Or you could use the owl statues, but those only work once (if you save nd quit w the statue, then return then quit, it wont bring you back to the statue.)
So the only way to deal with the moon is playing the song of time, or playing the oath to order after u have all of the boss remains during the little skull kid fight thing. After you beat a temple, you should play the song of time to return to day 1 so you can save that boss remains