A few minutes into the movie, we encounter the antagonists proper. And in that first scene, one of them kills a dog: It was barking at her.
It is then the movie announces it isn't really about the question of the DC-verse's fundamentally nonsensical no-kill rule. It is not here to present a thoughtful answer to the problem of the Joker Immunity
. Superman Vs The Elite
is plain old good-vs-bad, contained in a wild swing at a defenceless prop, stuffed full of straw.
Nowhere in the movie does Superman answer the simple math: Killing one superpowered, homocidal monster spares a dozen innocent bystanders the next time he escapes prison - however unlawful that execution may be.
So Manchester Black kills Atomic Skull instead.
Superman's only answer is this: To put on a Joker Grin and a savage, televised beatdown on Black, apparently killing the rest of The Elite in front of millions of viewers. Then, when people tell him not to kill Black, he says: "They saw the ugliness of violence as a solution, and it frightened them". Even though "they" saw Black blow Atomic Skull's skull apart just the other day, and cheered.
Superman considers himself a teacher, one that knows better than the great majority of the citizens about whether it's wrong or right to kill mass-murdering monsters. Incidentally, those citizens are the ones doing all the dying at the hands of said monsters in this movie. Manchester Black says it: "You don't have a clue what it means to live in fear, do you?"
The worst part is it didn't have to be this way: Superman doesn't have to be uncool. He could be interesting simply by being The iconic superpowered hero, and generally supposed to be the most powerful one in his universe. Even without the Justice League, he would have immense political and social influence. As Louis says, "The world is always watching Superman."
And if that was true, the abovementioned "ugliness" finally becomes relevant: When Superman's public persona becomes too important for him to get his hands dirty, when there are others who can do those jobs, and when the justice system isn't too completely insane to ignore the unreliability of Cardboard Prisons
, then Superman at last has a good reason not to kill. Or something. But at least address the central question of your movie with a real argument.