I believe he should stay true to his honorable self

There are times I go back and forth on this, and in other alternate worlds, I don't even really hate that he takes Joker out, but in the mainline universe, I do prefer that he stays true to that rule.

No! It's one of his core elements.

To stop the Joker for good, that is the only way and Batman knows that.

He would have to change his ways to stop him.

Batman's no kill rule is a product of editorial mandate from decades ago and, even at that, isn't something he's strictly adhered to over the years. It's actually one of the things that makes Batman interesting in that there is this ideal and then there's the flawed implementation of it. I don't think we as readers give that enough credit.

The bigger question really should be would killing the Joker actually accomplish anything -- and I don't think it would. Batman letting the Joker live and seeking other ways to stop him is more a mechanism of understanding that killing him would only make him more powerful by extension (meaning the damage to how people perceive Batman, people who worship Joker taking up his cause, etc.) than Batman's "no kill" rule.

I always thought he should, but now I think he should stick to his strict set of rules. There would likely be fallout if he did start now.

It's a great debate because does one truly exist without the other? Now, that's not exactly fair to the people of Gotham, but I don't think Batman is as effective if he doesn't have that White Whale to chase.