A very symbolic dream indeed. But very difficult to interpret since it is very personal to you. 
I do also believe that the moon represent that deeper part of you, emotions and intuition. Perhaps it is trying to tell you to get in touch with that other side of you so that you live completely in balance. My guess, the sun represented the physical aspect where you put most of your focus towards and the moon your innerself/emotions. Hard to say really. The false moon could also be a reflection of the other artifical sun. Just like how the IRL moon reflects the lights of the sun.
I just noticed it has been a couple years since this topic was first posted lol. Hopefully you’ve already found your answer.