I have what might be considered unpopular and not politically correct opinions on a couple points you made.
I don't think you're going to get a lot of traction on your request for adult supervision. If the kid is running around with a BB gun unsupervised it's unlikely it will be taken away and only used with his parents around. I'm not even sure you can expect that of a 13 year old, when his parents have already given him free reign. I can remember when my friend got his first bb gun, we shot everything that moved and lots of things that didn't. We didn't mean any real harm, though.
About shooting at moose and other wildlife. I don't think you're going to get any satisfaction there either. it's unlikely if he ever gets it in his head to shoot at a moose, that you or his parents are going to be able to stop him.
Now shooting at you, your house, your dog, your stuff. That's a different story. I'm extra sensitive to getting on the bad side of the neighbor kids. Especially the 13-17 age ones. In our last house, I looked a little too long (in their opinion) at a small group of them at our corner one day. My reward... Four smashed car windows over the next 3 weeks. Not much I could do to stop it. Thankfully I eventually figured out who it was and threatened the parent that I had video footage (I lied) that I would take to the police if it didn't stop. the blackmail worked. I never saw those kids again.
Oh, to think of all the things I would have liked to do to those little brats... But in the end it was best to de-escalate.
I figure if you push it with the parents, get his bb gun taken away, you better hunker in for war. I doubt a 13 year old has much else going on in his life except to make you miserable. And it's not like you can kick his butt, or you'll be the one going to jail. Plus who wants that grief from their neighbors.
If I were you I'd have a talk with the kid. I'd be nice, but I'd tell him outright, you would really like it if he would shoot someplace else. I wouldn't mention the ordinance or talking to his parents. Treat him like an adult and he might just act like one.
if you think he is shooting at animals, I'd mention that a good hunter avoids being cruel to animals. Shooting an animal with something underpowered might at worst get you killed and at best hurt the animal for no good reason. Hell, tell him (or make up) a scary-ass bear hunting story, so he knows you know your way around guns and maybe looks up to you as a fledgling shooter.
And, uh you might want to buy some surveillance cameras.