As for the trueness to the source material. That's almost an irrelevant critique now. Everything has been "reimagined" or is part of an alternate continuity.
In and of itself, no. There's nothing wrong with reinterpreting a work of fiction, and Batman has certainly been reinterpreted many ways over the decades. It's just that when the film came out, lots of people claimed it was more faithful and true to what Batman was than previous screen adaptations had been, and that's simply incorrect. It was superficially darker and grittier, like the Batman comics of the day, but it didn't really have much else in common with them.
The idea doesn't work realistically, because, sooner or later, Batman would kill someone by accident.
Accidents are always possible, but that doesn't mean it's unrealistic for a character to strive
to avoid taking life if at all possible. After all, that's kind of what police do in real life -- when they do use lethal force, it's supposed to be an absolute last resort when all other options have failed, and there's a lot of effort to develop new less-lethal weapons.
Indeed, it's actually more
plausible for a civilian vigilante to strive for nonlethality, because if he does kill, then he won't have the legal protection and support that a state actor like a police officer or FBI agent would have, and would thus be vulnerable to homicide prosecution or wrongful-death lawsuits. Look at the "real-life superhero" Phoenix Jones and how quick the Seattle police were to crack down on him just for using pepper spray to break up a heated argument. The police tolerating and cooperating with a nonlethal crimefighter is barely plausible; the police tolerating and cooperating with a deadly vigilante is completely beyond belief.
I wasn't saying to turn Batman into a "Punisher" type character. But a vigilante who's taking on rooms full of violent criminals wouldn't be able to worry about defending himself and simultaneously making sure he's not killing a single person there. He'd strive to avoid deliberate killing, but would do it in the heat of battle if he had to.
Heck, Nolan's Batman does this anyway in TDK with Harvey Dent, even though one of the issues is that he WON'T kill.
It's sort of like "uh, if you could do THAT, why did you worry so much about killing the Joker?" You could have saved a lot of lives earlier.