Here's my opinion on this.
Firstly it appears to be a pastiche composition possibly by Melville himself of bits and pieces he has heard/read.

The first quatrain is as you say. I would guess the next quatrain is Melville's, and the last couplet is a variant of a very old song represented on the folk scene by 'The Bonny Ship the Diamond' a Scots variant, (See DT) but there is a London antecedent from the 17th century if my memory serves me right.

I have opinions to offer on how the song evolved but they are only opinions.