I think for me the priority is that he's well masked and manipulative, that he has built up a reputation in the city and gained the kind of power that only comes from controlling public perception.
If Lex doesn't care about how he's perceived by the world, if he doesn't care about being top dog, about getting away scott free with all the awful things he does, because no one can touch him... then his rivalry with superman loses something for me.
It can still be done... I just like it much more when Lex sees it as a battle for the top spot, rather than just an anti alien grudge or jealousy alone. I want Lex to be both jealous AND competitive/challenged.