OK, so far I've discovered that the song is "Choo'n Gum" by Vic Mizzy and Mann Curtis, recorded by, among others, the Andrews Sisters, Dean Martin, Teresa Brewer, and Maria Muldaur. I do remember that my dad used that "odd" pronunciation of "chewing gum" when he sang the lines. I haven't found an MP3 yet, but, assuming that the tune is the same, Mizzy must have picked it up from somewhere, since Gershwin used it in 1928. Curiouser and curiouser...

Aloha,
Mark