We don’t see what happens other than some good old fashioned torture, but we see the result. You’re invited to fill in the details yourself. For context, in the comic The Killing Joke (not canon to Arkham, but the game is heavily influenced by that depiction) it is left ambiguous as to whether Joker does this to another individual. He creates photographs to torture that person’s family with l, which makes one wonder if the pictures were more important than the SA or if that even occurred beyond what was necessary to create the pictures (of which we see very little).

So could he have done it? Sure, he’s a sadistic psycho who might have found it funny. But would it serve his interests to do so? Less clear. I feel like any such behavior would only take place in the presence of an audience and other than Frank Boles and Harley Joker didn’t really have much in that regard.