i don't think lex cared. because if batman killed superman then he wasn't all powerful, and if he killed batman then he wasn't all good because he took "justice" into his own hands and became an executioner. doomsday was a last resort that if superman survives batman he could kill him.
This is quite how I see it too.
Lex wants to get rid of the meta-humans/gods that are on Earth.
Discovering that the kryptonite could kill Superman, Lex decides to start with him and to use Batman to make the dirty job.
Knowing that this one has lost people in the battle of Metropolis and that he's in the path of a radicalization, he makes sure to set up Superman as a threat to push him to the confrontation.
On the other side, he tricks Superman to go after him by killing the prisoners having the bat-brand. This way, he makes even Batman angrier knowing that this one won't tolerate any kind of control, let alone from someone he considers responsible of personal loss.
Bringing the kryptonite to Gotham, Lex "helps" Batman to have the rock and then kills an ex-employee of Wayne Industry in martyr to completely settled his rage.
Finally, he forces Superman to go to the battle and hopes to see the bat succeed.
Doomsday is an alternative born with the opportunity to have had access to the kryptonian ship. Thinking he can control it by mixing his blood with the creature, it was a last resort for Lex to kill Superman and, I think, the way to kill the rest of the meta-humans/gods he was tracking.