The whole motivations, plot, and plan exhaustively explained here:
However, with respect to specifically Doomsday, first you correctly conclude Lex didn't care. We have so many proofs of that. The timing of Doomsday's creation, Lex sticking around rather than retreating, Lex engaging in Communion, Lex intending to tell Superman Martha was dead even before Doomsday was done, Lex confession to most of his machinations to Superman, etc. Even if Doomsday never arrived, Lex had already brought down a storm of consequences through his actions which indicated he didn't care.
If we have to briefly sum up what Lex wants, it's to show the world that there is no benevolent higher power, that God is dead, but if there was a supreme power that it would be malevolent, cruel, corruptible, evil, and destructive. So to him, Superman is a fraud to be exposed, and it has to be done in such a way that Superman isn't simply martyred and replaced by another virtuous metahuman ("gods among men") forcing him to go through this all again.
He go through the story trying to demonize Superman so he can slay the symbol AND the man.
When Lex enters the Scout Ship he asks it two rational questions:
How do I kill Superman? (He is no longer in control of the Kryptonite and needs a backup is Batman doesn't do it.) The ship teaches him about Doomsday.
Any others like Superman? (Lex is wants to deter the metahumans from replacing Superman. There's no point to his plans if the cosmos is crawling with similarly-powered paragons waiting in the wings to replace Superman and restart his entire problem. Lex asks, "What's the most powerful thing in the cosmos which might come to our little blue planet?") The ship teaches him about Darkseid.
Between these two lessons, Lex learns that he was right and that they're all living on borrowed time. He wins no matter what because of Darkseid, but before that he wants to make his statement know to the world.
Doomsday is the perfect expression of that statement. Lex doesn't need to demonize Doomsday. Lex doesn't need to explain or put Doomsday under duress. Doomsday is leagues more powerful than Superman and more powerful than any metahuman on Earth. Doomsday is the devil incarnate and proves exactly what a supreme power on the planet would be and that god is dead.
So everything shifts to making Doomsday exist, which is why the plan becomes so transparent and unhinged. If Lex really just wanted Superman dead and if there was time to do it, Lex has proven he's willing to use proxies and strike from the shadows. He doesn't do this in the final act because Superman is a dead man walking and this is his last chance confront the alien who caused his existential crisis and say how he feels to its face.
Beyond that Lex doesn't care. Not because he's crazy but because he's transcended mere villainy. He's gone through all the religious rites and is now a supervillain. He's been confirmed by the ship, baptized by its waters, learned its teachings, he makes a blood sacrifice, communes, and in the end with his head shaven in devotion, he makes a prophecy. Lex is devout and devoted and as willing to die for his beliefs as Superman ultimately does for his. He is Superman's match and nemesis.