You're right! "Moustache Pete" in the Thirties and Forties connoted an old-time Sicilian, dressed and wearing a drooping moustache typical of 19th Century Sicily. Americanized gangsters like Charlie Luciano, Frank Costello and Joe Adonis held them in contempt, and applied that monicker in a derogatory way.

Ironically, the last two "Moustache Petes" in NYC, Joe Massaria and Salvatore Maranzano, who fought each other in famous Castellemmarese War of 1930-31, were clean-shaven.