Even outside the Dark multiverse Both of these iconic characters have killed The Joker. superman kills the joker In injustice and batman has canonically killed The Joker in at least 2 comics. I will try to keep spoilers to a minimum but the following comics have been around for 10-20 years.
batman has killed the joker in a dream sequence superman has Injustice (DC UNIVERSE INFINITE)
in the 3 book series Justice League of America: The Nail (DC UNIVERSE INFINITE)
In Batman’s case he kills the joker, batman turns himself in to the police.
Superman kills the joker in injustice, Superman becomes a tyrant. Now admittedly superman was lead down the path of the tyrant by Wonder Woman in injustice. (that is so wrong on so many levels)
However in JLA: The Nail batman finds his rouges all killed, watches The Joker knock Catwoman out with a Kryptonian energy weapon, then the Joker forces Batman to watch as he tortured to death Robin and Batgirl. Arkham blows up batman nearly escapes carrying Selina. falls into a catatonic state, comes out of it, and once the big bad is defeated batman turns himself in and is acquitted. In the dream sequence in injustice, Superman saves Lois, joker says asks if batman will get jealous if he goes after superman again, as a result, he breaks jokers neck in the Batmobile and turns himself in.
superman in Injustice just goes on as business as usual after everything that happens. then just keeps making one bad desison after another ultimately culminating in the death of all but four members of the bat family (Batman Catwoman, Damian Oracle) and batman is still unwilling to kill superman despite all he lost in a war with superman.
what this says about the two is this:
Superman is an Angel one step from falling from grace, Batman is a Demon who is seeking redemption. an angle can easily fall from grace and superman falls easily. It is almost impossible for a demon to go to grace, Yet Batman does not kill despite facing true evil like the Joker and Victor Z.
However, I wonder what others think of this idea.