I understand that by some point in the 19th century it was a racist term referring to black children, but was this always the meaning? I was recently reminded of a lovely little ditty from the Golden Age of Piracy (I believe I first read it in Stephen Talty's Empire of Blue Water), which uses the term in a way which could be consistent with the more modern usage, but is open enough that it could easily mean something else. The song:
Him cheat him friend of his last guinea
Him kill both friar and priest- oh dear!
Him cut de t'roat of picaninny
Bloody, bloody, buccaneer!
So, was this just about buccaneers killing children, or is it something else? As well, if the basic meaning was unchanged, was it always a specifically derogatory term?
According to the Oxford English Dictionary, it appears that the term you think it means, small black child, is the one being applied here. The term appears to have origins around the mid seventeenth century. While the word could be used just to mean small or tiny, this poem is using it as a noun, so it means the black child.
Also, I've yet to see a date for the poem you refer to - so I wouldn't say it's from the period just yet. It might be a later invention. Plenty of songs are from much later that the period it talks about. Many websites list it and never mention the song/ballad's origins.