I'll argue the former since I completely disagree. One, his moral code on not killing has nothing to do with public perception, so that I'm going to toss aside is irrelevant. Why would it be questionable in realism for Batman not to kill the Joker? In reality the Joker would never escape, time and time again. Our society has that moral code right now, that killing someone is wrong, even deranged madmen aren't killed because of this moral stance. Batman is a vigilante to protect the people of Gotham. And part of that is to prove and show that the rule of law is still something, and means something. It's why he works closely with men like Gordon, and Harvey Dent before his accident. If he were to kill people that's a break in the trust of the law.
Now let's take this to a look at the world these movies have created. In this world were batman is perfectly okay killing, who has he killed? Thugs of Lex luthor whose crimes were kidnapping (a crime not punishable with the death sentence) and theft (another crime not punishable with death). But he doesn't kill Deadshot, a well known assassin, and the Joker, a psychotic killer. This is a world where Batman who is willing to kill Superman before Superman commits a crime just to be sure, but doesn't kill Deadshot or the Joker. Sure, it's realistic.