Quoting from blackface minstrel-'Ethiopian' routines or descriptions of them does not help your case that 'bulgine' comes from Black speech.
The chantey is based on the minstrel routines; the similarities make that obvious; but minstrel speech must not be equated with Black dialects of the time.

The term is derived from the patents of Edward Bull, the Bull engine, shortened to bulgine; so derived in the OED.. In the 1840s, the ability to increase the pressure of small steam engines to over 60 lb/sq. in. without explosion (or so the manufacturers said) made these engines ubiquitous in dockyard, ship, fire, mining, construction and other pursuits.

The chantey is based on the minstrel routines; the similarities seem ample and that is not denied. Sailors coming ashore in the major ports sought entertainment in the minstrel and music halls, and picked up the songs that appealed to them.

I have recently acquired a set of Doten's Journals, covering the development of California in the period 1849-c. 1900, published by the University of Nevada Press. I look forward to reading this important historical work; not about the sea, but about life in California and Nevada during the Gold Rush and into the period when California was joined by rail and steam to the rest of the Union. Doten describes a bulgine exploding, in 1849.
The steam engine, or bulgine, was a contributor to the building and exploitation of the West.

Bulgine as a word in the chanteys, as you say, is not important, but you claim Black origin without evidence, as you also seem to suggest that a chantey preserved in the Caribbean or Demarara area is necessarily of Black origin.