has lyrics that might fit but says to put it to a regular tune "Santa Claus is Coming to Town"...
Didn't see a place where they had it actually played.
Those lyrics are a lot closer, but in the song I am looking for, Santa never shot the reindeer, they were part of his team.
Santa never shot innocents, just those who had abused him.
It played on the radio, in at least Madison, Wisconsin, probably on WKOW.