FWIW, Harry Belafonte, while born in Harlem, was the son of Melvine (née Love) – a housekeeper of Jamaican descent – and Harold George Bellanfanti, Sr., from Martinique.
From 1932 to 1940, he lived with his grandmother in her native country of Jamaica.
His Caribbean dialects were thus not mere imitations but echoes of the speech he heard from his own parents as a child.
http://en.wikipedia.org/wiki/Harry_belafonte

I have no problem with songs being sung using or approximating the pronunciation or dialect in which they were originally written or sung.
My issue is how, when a dialect or language is transliterated using English letters (instead of phonetic characters), it often leads to gross distortions of the actual sounds.

So, if you read an old "minstrel song" as "Heb'n, heb'n, ebrybody talkin' 'bout heb'n ain' goin' dere, g'wan ta wa'k all ober God's heb'n," and you try to reconstruct the dialect from that transliteration, you will probably really exaggerate the "bad English" and not sound much like the actual dialect anyway.

I'll bet this goes for a Caribbean dialect too.