This whole elaborate thing would have been a lot to try and fit into the film, and there is a nice little Easter egg to the “Aussie Lex” in the movie, but removing it just to give a weaselly, smarmy version of Lex rather than a cool and collected one feels like a bad calculation.
Putting Lex on the scene at every stage of the story, as if he were somehow an essential element of the Doomsday tale, is equally baffling: having LexCorp divers inspecting the asteroid that brings Doomsday to Earth becuase on a monitor he thinks the rock “looks pregnant,” and then having him jump into a suit of Kryptonite-powered armor to try and take down the monster…meh. We could do without any of that. Lex is a great character, but “The Death of Superman” was never much of a Lex Luthor story.
Moreover, for him to say that Doomsday killing Superman “would save me a lot of trouble” is a bit out of character…maybe not for Lex as a whole, but certainly for this storyline. In his most famous scene in “Funeral For a Friend,” Lex gloats over the coffin, exalting in having outlived his nemesis…but during the battle itself? He didn’t want any hand but his own to be responsible for putting Superman in the ground.