This referes to the feast of the Epiphany, though, which is 12 nights after Christmas...
That's why I say it would've worked better for me, yes.
That would "require" a timeline shift for it to start on Christmas run until the Ephinany.
Although I don't think anyone really knows what the 12 days of Christmas are anymore.
We have a thing called "Three Kings" (aka. "three wise men") in Poland, that falls on Jan 6th. If My Math Is Correct™, there's 12 days between the Christmas day (Dec 25th) and Jan 6th, so maybe the song is about this period?