This fairly comprehensive list of songs and poems printed by (of all people) John Raw in 1818 makes no mention of the song that I could find, though perhaps someone more familiar with the area could find something useful in it. I will mention that the OCR is fairly crude and not very good, so if you have the time it may well be worth a closer look.

There appears to be a copy of the physical version of the song at the Australian National Maritime Museum, so if you know someone who will be by there they could check.