The same version of Jingle Bells/Batman smells and the batmobile losing its wheel is found in Barbara Michels & Bettye White's 1983 book "Apples On A Stick, the Folklore of Black Children" {from Houston, Texas}

I have also collected that same exact rhyme from Pittsburgh area children & youth. Another Batman rhyme that I collected from children in the 'Greater Pittsburgh area' is this one that alludes to Batman & Robins' allegedly homosexual relationship:

Batman and Robin flyin in the air.
Batman lost his underwear.
Batman say "Me don't care.
Robin'll buy me another pair.