Although the most promising thing now seems to be to find The Book, I have some additional thoughts.

I’m taking Tommy W’s memory of The Book with a grain of salt, in case looking for a book with all those characteristics fails. (He might have the memory of two different books mixed up.) What should the poem in the book look like? Will it have verses that obviously match “Wellerman”? Will it have the word “Weller” in it? Or will it just vaguely resemble the shape of the song?

On the chance that the resemblance between the poem and the resultant song is tenuous, my thought is to look at the poetic form. In any case, we need more clues to search for The Book.

So.: “Wellerman” has a pretty distinct AAAB rhyme pattern.

She had not been two weeks from shore
When down on her a right whale bore
The captain called all hands and swore
He'd take that whale in tow

How common is this? Probably others can think of many more examples than I, but I’d still guess they are few enough to call it distinctive.

On one hand, there’s “Golden Vanity.” Yet the other that comes to mind is “King of the Cannibal Islands” (dating from at least 1827)

Oh have you heard the news of late,
About a mighty king so great
If you have not 'tis in my pate,
The king of the Cannibal Islands

In searching for candidates for The Book, two of my note were: The Whalers, from the journals of a surgeon on a whaler (c.1837-1846), Félix Maynard, edited by Alexandre Dumas, and translated by F.W. Reed, 1937; and Pioneering in South Otago, by Fred Waite, 1948.

In the Introduction to The Whalers, Johannes C. Andersen quotes “the first two stanzas” from a whalermen’s song from the barque Magnet, though its source is not indicated. Andersen quotes the same stanzas in Place-names of Banks Peninsula: A Topographical History (1927), that time giving the source as M. G. Thomson, “A preliminary paper to the history of Otago,” 1886.

Along the coast the Magnet came,
With Captain Bruce—a man of fame,
But in his face there is no shame—
On the beautiful coast of New Zealand

Mr. Wiltshire sold to "Bloody Jack"
Two hundred of flour tied in a sack:
and the Maori carried it all on his back—
On the beautiful coast of New Zealand

At least one other work, Maori Music, quotes the same.

The actual source information is slightly different from what Andersen said. It is “A Preliminary Page to the history of Otago,” Evening Star [Otago], 25 October 1884.
https://paperspast.natlib.govt.nz/newspapers/ESD18841025.2.35.5?end_date=25-10-1

The newspaper article gives the 2 additional stanzas:

Waikouaiti and Molyneux,
Tautuku and Otago, too—
If you do not want to be duped by a Jew—
Come to the beautiful coast of New Zealand.

Peter Shavatt has a shocking bad hat,
And old John Hughes, with his shocking bad shoes;
But for all of that they are having a chat—
On the beautiful coast of New Zealand.

Pioneering in South Otago quotes the third and fourth stanzas.

This is the song mentioned by Bob in February.

The New Zealand Folk Song website takes this song to have been a parody of “King of the Cannibal Islands.”

https://www.folksong.org.nz/beautiful_coast_of_NZ/index.html

Well, through an intermediary parody about Australia:

Upon the voyage the ship was lost.
In wretched plight I reached the coast,
And was very nigh being made a roast
By the savages of Australia.

https://books.google.com/books?id=g5EYAAAAYAAJ&pg=PA20&dq=“In+wretched+plight+I+


Many parodies were made of “Cannibal Islands,” and it wouldn’t be surprising that foreign visitors to NZ/Australia had the song in mind.

I’m not making a real argument here. I’m suggesting that we keep our eyes open for verse that conforms to this pattern.