The Killing Joke ending

I rewatched Battle of the Bastards and a thought came to my mind. It concerns the ending of Alan Moore's The Killing Joke. 

At the final moments of the book, we are shown Batman and Joker sharing laughter after the Joker tells a joke about two patients escaping an asylum. And then the laughter of the two abruptly ended. A lot of people suggest that maybe Batman already snapped and ultimately killed his foe by chocking him or breaking his neck.

But it doesn't make any sense if he really killed the Joker. Although there's no questioning that what he did would be for the greater good and Batman had already enough cause to kill him, it still doesn't make any sense. It mirrors Jon Snow's cause in killing Ramsay Bolton. When he saw Sansa, Jon knows that she has the right in deciding Ramsay's fate because she is the one raped repeatedly by him. Gordon has the right in judging the Joker's fate. It's his daughter that was paralyzed and (possibly) raped. It's him that was also sexually assaulted by the Joker and shown pictures of his naked daughter. Yet he still insists in not killing the Joker because like Batman, Gordon has principles and he chooses to uphold them. Before Bats confronted the Joker, Gordon says to Batman that he has to show the Joker that their way works.

That's why Batman killing the Joker doesn't fit his persona and the situation presented by the book. In doing so, he will fail, not only his principles, but also his friendship with Gordon. He should respect Gordon's decision in letting the Joker live, because Gordon has every right in choosing what he will do to the Joker, no matter how bitter that sounds to Batman.