If it’s “Joker got away,” you’ll have to change the second line (using dreaded inverted word order–you know, when verbs misplaced are…) to “Robin eggs did lay.”
I also am from the 70s and I always sang, “Joker got away”
But I can see someone trying to rhyme egg, so “…broke a leg” would make it rhyme
The same for me, circa 1990 (Virginia). The implication was that the Joker took up a new pastime that indicated a renunciation of sorts of his former life.
— for me when I was in elementary (starting 1990 in FL) it was ‘laid an egg’ I don’t remember if it changed any when we moved to NC. But while reading this I kept thinking that there was an extended version with something about ‘dropped a gun’ I think that was with the part of the leg breaking, but don’t quote me.
The OP’s version was also my version.
“Broke it’s wheel” IIRC. Early eighties, I guess.
No, no, no. It was Joker broke his leg! At least that’s what I always heard when I was a kid in Central NY in the early 70’s.
Really? I always thought of it as, “ha ha, big scary bad guy is doing something silly and girly.”
An Australian version from the late 70’s early 80’s;
Jingle Bells, Batman Smells,
Robin laid an egg,
Wonder Woman lost her bra,
Flying T.A.A.
So apparently Aussie kids are less about the violence and more about the sex 
Oh and TAA was a well known Australian Airline of the time (Which has since gone bust)
cough
Early 80’s Michigan
Jingle bells
Batman smells
Robin laid an egg
Batmobile
Lost a wheel
And Joker got away
though for some reason in first grade some kids thought it was really funny to add onto the ending
And Joker got away
With his sexy legs
Keep in mind, none of us even really knew what sexy meant, but here we are three decades later and I still think that line every time I hear the song, even if I haven’t sung it out loud since then.
“Joker got away” is how I learned it in the Bronx in the early 70’s, but this is what I’m going to sing from now on.
Somebody upthread grew up with the Commisioner version and only heard “Joker” on the Simpsons. Odd, since I first recall seeing “Commisioner” in a Groening “Life in Hell” strip titled “kids’ greatest hits”.
Child of the seventies here. The Joker got away for me, too.
Regarding the “Robin laid an egg.” line:
This is a very clever play on words with at least three levels.
1 - Robin is Batman’s sidekick and a kind of bird.
2 - Birds lay eggs. (Robin, as in Dick Greyson, laid an actual egg. Ha-ha)
3 - Laying an egg is yet another euphemism for farting. (Robin, as in Dick Greyson, farted. Farts are funny.)
But either way, that would be good for Batman. I have to say, given a choice between the Joker doing ballet and peeing bees, I’d take the bees.
Never heard about anything happening to the Commissioner. On our playground, the Joker got away. Which didn’t rhyme, but that didn’t bother us.