TL;DR: Kayn and Rhaast should team up because it's the best way to progress both their stories in a satisfying manner.
If the lore progresses and Kayn wins, that's going to end up understandably disappointing to Rhaast fans. The guy has barely any backstory beyond a handful of vague mentions and contradicting implications. Kayn "killing" him (even if he'd be back upon Kayn's death) takes him out of the lore with no answers or chance for his character to progress.
If the lore progresses and Rhaast wins, that's going to end up understandably disappointing to Kayn fans. Sasuke Uchiha jokes aside, Kayn was a child soldier who probably has very few friends by virtue of being Noxian and surpassing basically everyone in his Order. He undertakes the ultimate challenge, risking it all to prove himself worthy....and then his story just stops because Rhaast starts a murder party.
No matter which one wins, the other becomes a shaggy dog story. To me, both of those outcomes are disappointing.
The best outcome for everyone involved is for them to *both* win. They have a decent amount of stuff in common to bond over. They were both denied personhood by those who wanted to use them as weapons, they both see *themselves* as weapons as a result, they both are very good at killing, and so on.
They could switch off on who's using their now-shared body to preserve the in-game mechanic of going either Darkin or Shadow Assassin. Maybe it's not "realistic" or whatever to have a happy ending, but I can think of no better conclusion to a tale of two edgelords than the power of friendship.