Idle speculation is only worth so much, but if I may be allowed some idea chat:

Toward the end of hunting a theoretical source text, my current thought is to expect something like this—

Two stanzas, which suggested a pattern.

First stanza
Begins with some variation on "a ship that went to sea / put to sea / was sailing on the sea." Its second line tells the name of the ship, something that rhymes with "sea." This is a common way to begin a "Golden Vanity" rendition, though it narrows things somewhat by eliminating version that open by saying e.g. "there once was a skipper who was boasting on the quay."

Second stanza
Begins with some form of saying that "they/she/ the ship" "had not been / hadn't been" sailing for but [X length of time]. This narrows it down to another pattern that is consistent in Golden Vanity variations.

"Wellerman" follows both of these patterns for its first two stanzas.

It could be narrowed further by looking for another pattern, though less common, where the variation forms three rhyming lines, followed by the "Lowlands." The more frequently encountered (in my experience) rhyme pattern has two rhyming lines followed by two lines about Lowlands.

To reiterate, "Wellerman" has 1) a stanza about a ship going to sea, followed by a rhyming name of the ship 2) a stanza about the length of time at sea with the characteristic grammar "had not been" 3) sets of three rhyming lines.

The third characteristic is a wild card though. The writer(s) of "Wellerman" may have found the sample fragment in a two-rhyming-lines form, then went to references other versions of Golden Vanity, found three-line forms, and adopted that idea.

Another helpful earmark could be whether the same text elsewhere plants the idea of a "blow, my bully boys, blow" chorus for mariners' songs. That's another wild card since it's a familiar trope anyway.

A an example that fits some, not all, of these criteria, Verrill's _The Real Story of a Whaler_ (1916) begins a chapter with a stanza that, though we recognize it as a verse of Golden Vanity, could just as well appear as a stanza of poetry:

There once was a ship in the northern sea,
And the name of the ship was the Green Willow Tree.
As we sailed in the lowlands, lies so low,
And oh, we sailed in the lowlands O.

Elsewhere in the book are quotes two variations of "Blow, my bully boys, blow."

The text minimally includes other features described by Tommy Wood: stories about whaling and illustrations.

It lacks my hoped for criterion of a second stanza and three rhyming lines (each of which, again, could be gathered by cross checking other sources for Golden Vanity). Yet it makes up for this with a poetic rhythm that fits "Wellerman" perfectly.

I think it may be useful to search deliberately with variations on the formulation "had not been."

More idly, some collected versions of Golden Vanity use a phrase describing how the boy (who went into the water and hoped to drill a hole in the opposing ship) "bowed down," which reminds me of the "Wellerman" line about "her bow dipped down."