I too have read - in more than one place, I believe, but it's been years - that Jeannette and Isabelle (yes, Isabelle, not Isabella) were the daughters of the man who wrote the song. The song history that I read said that villagers used to have live "manger scenes" in the town square, and that the father was calling his daughters to come see the real-life baby in this tableau, with his mother (and, no doubt, a cow, sheep, real-life "Joseph," etc.).

If this account is true, then it makes perfect sense that the baby is "so white" with "rosy" cheeks, because it would have been European (French) actors in the tableau.

But Wikipedia -- that ever reliable, never changing source ; D - currently has a different story.   What Joe posted sounds right: we may never know the history of this song for sure, any more than we know the history of "Greensleeves.