After watching batman throw around explosives left and right I no longer believe batman doesn't use guns because of some sort of moral quandary but rather he just prefers the challenge.
He doesn't use guns because he specifically doesn't like them. If it were just about killing, he'd use nonlethal bullets. The no killing rule is a separate but related phenomenon.
As for explosives, he generally only uses flash bangs, but when he's fighting someone who can clearly take it (like Superman or Darkseid) or a robot that it's okay to kill, then he uses more lethal bombs.
You address it in the first part of your comment but if batman has no problem using lethal bombs than why not a gun? That is where I am at.
It has to be he specifically doesn't like guns. Other commentors say he doesn't like guns because his parents were shot. Which is fine but kinda misses the point imo. It's like he heard someone say, "guns don't kill people, people kill people." Then was like nuh-uh but never moved that logic to, I dont know, incendiary weapons. Anyway, I'm way out in the weeds here.
The man threw a batarang into the barrel of a gun. Surely he could master non-lethal force with a hand gun or something.
A batman that uses guns is not as entertaining. I can agree on that much.
Yeah, he specifically doesn't like guns, and it's because his parents were shot. Other weapons are fine. Even a grappling gun is fine. A gun that shoots bullets is not fine because it reminds him of how his parents died.