Lunar synchronus orbit may not even exist. The earth gravity disturbances for something that far away from the moon to have a 30 day orbit would likely wreck the orbit before it got through the first revolution
It would certainly have to have some means of propulsion to remain in moon orbit for any extended period of time.
Can you expand on “lunar orbit may not exist”?
That is the feeling I got also. The earth would exert a lot of influence on a lunar orbit, I would posit.