The song is "Jingle Bells". The first verse is about riding in a one-horse sleigh, and the horse is wearing bells. They always put bells on horses in the wintertime, in the nineteenth century, so people could hear the horses coming, because there was snow on the ground which muffled the sound of the horses's hooves, and there were no wheels hitting dry pavement. This was a safety device, much like a horn on a car, today.
So, a bobtail, refers to the bobtailed horse, the one horse, pulling the sleigh in the song, has a bobbed (cut short) tail, as you have figured out. Not a dog. Dogs don't pull sleighs in ordinary situations. (sleds in Alaska, but not big sleighs in the rest of the world - and I don't think they ever bobbed the tails on Huskies, anyway).
a sleighing song is the song you are singing while riding in a sleigh...or about such times, like, for instance. "JINGLE BELLS", we are referring to the song that we are singing...right in the song we are singing...
a sleigh song wouldn't work...just wouldn't...it isn't about "sleighing", (the act of driving around town in a sleigh, which is what is going on in the song) nor does it fit the music.
BTW the jingling bells are a happy, bright sound, which is what the second line of the verse is referring to...how that happy sound makes people happy.