There's no such real word, but it is a corruption of another made-up word in a 1954 song called "The Letter" by Vernon Green of the R&B band, The Medallions.
Vernon said puppetutes was :"A term I coined to mean a secret paper-doll fantasy figure [thus puppet], who would be my everything and bear my children."
The old song is scrathy, but soulful and Miller mis-heard puppetutes as pompatous.
Actually, there's an American film called "Pompatous of Love" in which this line is a puzzle for the main character.