Well, it's about whaling, and the refrain is,

"Look chearful, my boys, let your hearts never fail,
Be brisk Harpenears when you strike in a whale."

Other than that, I don't see enough resemblance with "The Diamond" to call it the same song at all. Indeed, it bears a slightly greater resemblance to the later song that scholars, fans, and presumably nobody else, call "The Greenland Whale Fishery" solely because of that resemblance.

Melville's "Moby Dick" (1850) tantalizingly includes the lines,

"So be cheery, my lads, let your hearts never fail,
While the bold harpooneer is striking the whale!"

These are identified merely as being from a "Nantucket Song."

Perhaps Melville got the lines from J. Ross Browne's "Etchings of a Whaling Cruise" (1846), where they make up the refrain of a song ("Captain Bunker") that is an odd hybrid of "The Greenland Whale Fishery," "The Bonny Ship the Diamond," and "The Coast of Peru."