I grew up with the Adams/O'Neil era and I don't ever remember it being specifically stated that "Batman doesn't kill". It was a comic book like most all others, none of the super heroes did any killing really, not intentionally anyway. When did this whole "Batman code" start, because I don't remember it ever actually even being a thing.
Comics were for kids so the killing was no longer allowed as that came under scrutiny, he had to stop, Comics Code Authority probably sealed it later, and it just got to be part of the characters development well incorporated by the eighties, he would not use a gun, he would not kill. And this works because he is acting outside the law and is tolerated because he doesn’t kill, no guns means he has to be extremely trained and disciplined. And perhaps the gun would make it too easy for him to cross that line, he can’t become like those that take lives. It just works. He starts killing Gordon can’t work with him anymore.