Does Batman REALLY not want to kill the joker because of no kill rule?

I was doing a project at school about Under the red hood and I’m analyzing the scene where Jason asks Batman why he doesn’t kill the joker. And frankly, I think Batman’s “no kill rule” reason is bullshit, for the Joker at least.

I think the real reason Batman keeps the joker alive is because it keeps him whole as Batman. He doesn’t do it because he needs to keep staying as Batman, he’ll he even says it himself multiple times, how if there’s no Joker there’s no one for Batman to lock up. I think that’s the real reason he doesn’t kill Joker, it’s because he needs it to keep himself in check. It could honestly go for everybody else in his Rogues Gallery too.