Still haven't found the old song book I remember having it in, but James Leisy's Good Times Songbook (1974) has this:

Traditional, American

Oh, Mister Moon, moon, bright and shiny moon,
Won't you please shine down on me?
Oh, Mister Moon, moon, bright and shiny moon,
Won't you come from behind that tree?
Oh, my life's in danger but I'm scared to run.
There's a man behind me with a big shotgun,
Oh, Mister moon, moon, bright and shiny moon,
Won't you please shine down on,
Please shine down on,
Please shine down on me.

Then Leisy states it's a 'Traditional American barbershop quartet song'. He also states that some material was included in his earlier 'Folk Song Abecedary', doesn't say if this song was one of those.

I still have vague memories of a slightly different lyric, (I know we sang 'bright and silv'ry moon') and at least one other verse, will keep looking.