The entire sequence feels well done, altough a bit puzzling in its relation to the wider DCU in some respects, but I think I would have bought that Lex Luthor wouldn't use Superman's secret identity solely because it's been given to him as part of someone else's plan and it would make him a pawn.
It works, for me, in the view of Luthor as being motivated (in his hatred of Superman among other things) by some kind of unbending sense of pride (in himself and in humanity.)