Since this thread is live again, I might as well post what I've learned in the last three weeks. Which is mostly negative.

First off, the odds that the Reed book is the right one have increased, because the others I've seen are not it. I had real hope for L. S. Rickard, The Whaling Trade in Old New Zealand. I have not finished reading this, but I have zipped through the pages, and there is no poem printed, so it doesn't look as if it's in that one.

I haven't even started The New Zealand Journal... of John B. Williams, but again, flipping through the pages shows nothing.

There are two minor signs of authenticity in the poem. Rickard reports that the standard number of boats on a south seas whaler at this time was four, as in the song, and the earliest reference in any New Zealand or Australian historical dictionary to "billy" is to New Zealand, in 1839, where it is used to hold tea.