Here's a song that uses the word to refer to a blue eyed baby, presumably white.

Go to sleep my baby
Close your pretty blue eyes
angels are above you
looking at you dearie from the skies
Great big moon a shining
Stars begin to peep
time for litte pickaninnies
to go to sleep

If you realy want to sing "picaninnies" you could perhaps sing this song first with an explanation of the origins of the word.

Bert.