Here's more from W. C. Russell. SAILOR'S LANGUAGE (London, 1883)

//
...I quote these verses at length, as a fair sample of the sort of " growling " Jack puts into his songs. Unfortunately he is somewhat limited in melodies. Some of them are very plaintive, such as "The Plains of Mexico " and "Across the Western Ocean," and others have a merry, light-hearted go, such as "Run, let the bulljine run!" "Whisky, Johnny!" "Time for us to go," " I served my time in the Blackwall Line."
//

He cites SANTIANA, ACROSS THE WESTERN OCEAN, RUN LET THE BULGINE RUN, WHISKEY JOHNNY, "Time for us to go", BLACKBALL LINE.

That's the earliest mention I've seen in print so far of "Run Let the Bulljine Run," suggesting it may be a unique addition. But then "Time for us to go" is one of those "lost" chanties of Dana. If he knows his chanties, why does he have to throw in Dana's titles?

In the continuation of the passage, he uses the term 'chantey'. Had that spelling turned up yet?

//
But the lack of variety is no obstruction to the sailor's
poetical inspiration when he wants the " old man" to know
his private opinions without expressing them to his face, and
so the same "chantey," as the windlass or halliard chorus is
called, furnishes the music to as many various indignant
remonstrances as Jack can find injuries to sing about.
//


Finally, he quotes the Salt Horse rhyme:

//
The provisions have for years been a sore subject with the sailor.
His beef and pork have earned more abuse from him than any
other thing he goes to sea with. " What's for dinner to-day,
Bill ? " I remember hearing a sailor ask another. " Measles,"
was the answer, that being the man's name for the pork aboard
his vessel. " Old horse," is the sailor's term for his salt beef;
and some old rhymes perhaps explain the reason :—

"Between the main-mast and the pumps
There stands a cask of Irish junks;
And if you won't believe it true,
Look, and you'll see the hoof and shoe.
Salt horse, salt horse, what brought you here,
After carrying turf for so many a year,
From Bantry Bay to Ballyack,
Where you fell down and broke your back ?
With kicks, and thumps, and sore abuse,
You're salted down for sailor's use.
They eat your flesh and pick your bones,
Then throw you over to Davy Jones."
//