That's not true. Despite what certain movies have shown, for most of his career, Batman has been portrayed as adamantly opposed to killing. The early 1939-40 stories had him use lethal force, but that was soon enough retconned away and Batman was given an explicit policy against lethal force. And the versions that did show him as a killer did not make use of his aversion to guns. Indeed, both Burton and Snyder gave him a heavily armed Batmobile.
If anything, Batman's refusal to kill was established long before his aversion to guns. Batman was often shown as an expert marksman and student of firearms, and the '66 TV series occasionally showed Bruce Wayne going on hunting trips. I don't recall seeing an aversion to guns established as a character trait of Batman until the '70s at the earliest, more like the late '80s. ("There Is No Hope in Crime Alley" showed Batman outraged that a criminal would dare to draw a gun on him in Crime Alley on the anniversary of his parents' murder, but that seemed more about the circumstances than about guns in general. I think later writers may have been inspired by that line when they gave him a more general aversion to guns, though.)