So there are generally around 5-6 "big reasons" traditionally given for why Bruce doesn't kill his villains, many of which have been explored in comics and others of which have been discussed and debated ad nauseum around the internet. Here's the four I tend to find most compelling:
Bruce's moral code that prohibits killing is ultimately what separates him from those he fights against. He follows a very strict deontological viewpoint of "killing is wrong, regardless of intentions or consequences." Jason, by contrast, has a "ends justify the means" consequentialist mentality of "if you kill a criminal, you prevent more crime, so the killing is justified." The ethical dilemma surrounding this issue is that who are you, random quasi-legal vigilante on a self-imposed quest to end crime, to decide when you are or aren't preventing "more killing/crime" by killing a criminal? What makes Bruce qualified to determine who should get to live and die? He doesn't think he is, so he's simply said "I don't have that right. I'll instead work to save everyone, regardless of who they are/what they've done (to me or anyone else)."
Bruce's "if I start killing, I don't think I'd be able to stop" is less "murder is a potato chip" and more about the rationalization it would take to take that first step off the edge. Essentially, if he kills Joker, why not kill Two-Face? What makes killing Joker fundamentally different from killing Two-Face? From killing Penguin? From killing Harley, Ivy, Killer Croc, etc? Why is killing the Joker okay but killing say....Victor Zzasz isn't? When does a villain commit enough illegal and morally reprehensible acts that extrajudicial murder is an acceptable solution?
Part of the point and purpose of Bruce being Batman in addition to Bruce Wayne, CEO of Wayne Enterprises, is the flexibility he has in pursuing justice, rehabilitation, and re-education. The entire point of Batman is preventing the Wayne Family Tragedy™ from happening to anyone else. Bruce's entire mission in life is creating a world where "no more children lose their parents to some punk with a gun." Batman is supposed to prevent more children from becoming orphans, more wives from becoming widows, more husbands from becoming widowers, more parents from losing their children. What does killing do except perpetuate that cycle of violence and undermine his core mission?
How is Batman any different from the cops if he kills? Batman can't be a figure to inspire reform in the criminal justice system (and specifically the GCPD) if he kills, because how does that make Batman any better than the corrupt system he claims to want to make better? Batman killing doesn't inspire hope that there is a better way; it would just be an extension of how Gotham's "justice system" works anyway. By refusing to cross that line Batman as a symbol encourages Gotham to be better than they are.
As an expansion of #2, you could very reasonably make the point that "but it's the Joker! He's different!" But is he? Is he really? He's certainly done more permanent personal harm to the Batfamily than most other villains (Babs and Jason specifically), but what about Two-Face smashing Dick to pieces with a baseball bat in Robin: Year One? What about Black Mask torturing Steph to death during War Games? What about Shiva killing Cass in Batgirl (even if she brought her back)? What about Ra's nearly killing Tim in Red Robin? What about Talia murdering Damian by proxy (via Heretic) in Batman Inc.? What about Bane murdering Alfred in City of Bane? Where's the line?
Bruce has seen 3-4 people he's either legally or nominally responsible for die on his watch, the other 3 tortured to near-death conditions on multiple occasions, 1 permanently injured, and had a villain murder the man who raised him and leave his body for Bruce to find, and yet the only thing anyone really ever talks about re: Bruce killing is Jason and the Joker. If the line is torturing one of his kids (or those flying under his banner) half to death, by all rights he should have killed Two-Face and Black Mask for what they did to Dick and Steph. If it's killing a 'family' member, he should have killed Joker, Black Mask, Bane, and Talia (also Lex Luthor and...technically Jacob Kane, considering everyone thought Tim was dead).
It's not just the Joker at stake here: if we start saying "Batman should kill the Joker because of what he did to Jason [and everyone else]," you start getting into really thorny questions about well...a lot of Bruce's villains have done some ridiculously morally reprehensible stuff, including mass murder and irreparable personal harm to someone he considers family. What makes the Joker's mass murder different from Ivy's mass murder (and yes, canonically they are both mass murderers) that justifies the Joker's death but not Ivy's?
You can of course justify that by saying "but Ivy's not unredeemable! She can be reasoned with and rehabilitated!" but...what if Bruce had made the decision that she was unredeemable and worthy of death before her No Man's Land-era redemption arc kicked off? It loops back around to #1, that Bruce doesn't feel personally qualified to make that moral and ethical decision about who gets to live or die, because he thinks everyone has the personal capacity to change (and what right does he have to take that possibility away from someone?)
....also personally I think "they deserve to die" and "I deserve to kill them/let them die" aren't morally/ethically equivalent statements and Bruce can certainly think Joker deserves to die without also thinking he is morally allowed to kill him (or morally allowed to let someone else kill him when he has the ability to stop them).
