Well, I’d have the same sort of narrative but instead of Elijah having some weird nonsensical vendetta against Katherine for no reason; I’d have it come out that Elijah and Katherine have been soliciting and communicating for years behind Klaus’ back and that Elijah had been helping Katherine to evade Klaus for centuries. (Imagine Klaus probably found out a few years in but allowed it to continue as he knew one day he could use Katherine to manipulate Elijah or even vice versa)
Therefore there’d be none of Elijah saying he wants the old teenage pre vampire!Katherine back because he knows and loves Katherine for who she really is and is drawn to it. Although they do have a convo once Katherine snaps Elena’s neck about how Elena’s going through the same thing as Katherine and how she should show some compassion (because, just as with Klaus, Elijah is still slightly optimistic that Katherine can be a better person despite who she is) they argue and Elijah starts to reason that maybe Klaus is right (once he reveals that he knows about Katherine) and that he is just a fool whom Katherine is using.
Katherine denies it and tells him that she loves him but he rebuffs her. They have that scene where they meet up and Katherine gives Elijah the cure and says that it is a symbol of her trusting him the way she needs him to trust her.
When Katherine turns up in 4x20 to see Elijah and asks him to leave with her he accepts. He leaves Klaus a note and says that if he makes any attempt on Katherine’s life he will lose him forever. Somehow Elijah gets dragged back into the politics of New Orleans, on The Originals and refuses to tell Klaus where Katherine is when he asks. At some point, Katherine turns up, pissed that Elijah abandoned her for family drama. Klaus wants to kill her believing it to be fair game since there was an unwritten rule that if he was to refrain from killing Katherine then that would be on the grounds that he’d never have to look at her again.
But Elijah stands by his statement and promises that Klaus will lose his brother if Katherine is hurt. Kaliah are a couple on the Originals, cue Kat bickering with Rebekah and brOTPpting with Kol and Caroline and generally causing chaos and giving her immortal bae stress lines ever, ever after.