"My curly headed babby" was also among lullabies my mother sang. And if she thought being a "picaninni" was a shameful thing do you think she would have sung it to her child? Likewise my dad playing songs (or at least their tunes) like Ol' Man River, or 'Swannee Ribba'. And we sang along, doing the accent, which we believed to be correct. And they taught me something about the shameful way people had been treated in the past, (because I never experienced any abuse of people for their colour or religion. Yes jokes were made about these things, but we all suffered that, and were taught to disregard it ("sticks and stones..." teachers told us). There's been enough threads about real and assumed accents here for all to know that it's not done to abuse people.
Seems to me that those who find words offensive are just looking for problems.
The SITUATIONS that produced these songs were offensive, as were the people that brought them about; but we can't alter that.

Incidentally the "pale" is "the fence" (paling). A delightful if unintended reference to the OP's song. Let's not continue to build them.