Golden Age Batman used guns. And, like most Golden Age heroes, he was a lot more cavalier about indirectly killing villains (or letting them fall to their deaths). When Superman made a no-killing oath back in 1942, it was a big deal precisely because it was an exception. It wasn't until Silver Age that the "no kill" rule became one of the archetypal superhero rules. Since this Batman is based on the Golden Age Batman, I don't have a problem with it.
Of course, you are entitled to your opinion, and there is nothing wrong with being uncomfortable with Batman using guns. I was just trying to explain where I'm coming from.