Somewhere between (where he actually sits) is more interesting than both. He's clearly dealing with emotional trauma, but that doesn't mean he's not also still human. And trying to be more so.

I don't really like the take that it is solely Batman's responsibility to kill the Joker. Batman acts as a vigilante, and in order to not take actions that one cannot provide restitution for, he has a strict rule that he cannot kill. He enforces the law but he doesn't act like he's the judge, jury, or the executioner. He stops them and he lets the judicial system sentence them. The people he hands over the Joker to every single time has far more ample opportunity to kill him. The police, the judge deciding on capital punishment, prison guards, a random bystander with a gun. None of them carry the same extrajudicial responsibility that the Batman imposes on himself in order to remain accountable or at least not start taking charge of life and death.

The problem wouldn't be that the Batman kills the Joker. The problem would be when he starts deciding whether others are better off dead than alive.

Both are interesting and both are generally how Batman is. It depends on the run. Sometimes he's dark and grows into a paragon, and sometimes the reverse is true. Totally depends on the author.