The only time Batman fires his own pistol in the era is in Detective Comics #32, when he shoots two vampires with silver bullets. There is some other gunplay in comics of the era, but none of the shooting is done with a pistol Batman is carrying. In Batman #1 he shoots Dr. Hugo Strange's henchmen from the Batplane's machine gun. He is seen on the cover of Batman #15 shooting a machine gun in a Nazi based story, but it is a dream sequence. By the winter of 1940 it is openly stated that Batman doesn't "carry or intentionally kill with" guns in Batman #4, even though he is seen seen "trying to wing" a villain with a recovered gun in the comic. The popular (at least amongst nerd circles) story of the old-timey Batman who carried a gun doesn't really hold up. he had one on his belt in some early comics, and he shot some vampires with silver bullets once. That's it.
There are other examples of Batman using guns in later comics, but these examples are generally where he uses a gun on the scene to assist the plot. The later examples of Batman using guns do not consist of him killing anyone with a pistol he is carrying. More often he is challenged to a contest of target shooting, or he is firing the gun for comedic effect or some sort of cowboy themed activity. You can read a pretty complete take here.
The point of this blog is to summarize. If you truly summarize Batman's activity in the early comics, he never shoots any living creature with his own pistol. He shot criminals from his plane mounted machine gun once, shoots two undead vampires with his pistol once, and uses a recovered gun to shoot a criminal in the hand (similar to the Lone Ranger's signature move).
As much as he didn't kill people with his pistol, he did kill people in numerous other ways, see below.