I have to agree in getting the authorities involved but I’m not sure how much they can actually do. Unless you’re willing to pay for a lawyer and pursue this, there appears to be a problem with “burden of proof”. Right now, it’s your word against his. You can file a complaint but how do you prove anything? The child has a different story, the mother is defending the child, and you’ve got……… your word. Any witnesses? Any video? Any proof the pellet gun was loaded? They “poof” the same, even when unloaded.
I’m not defending this child or his actions but there’s been some “puffed chest” responses here and I really think a complaint file is all you can do legally, but is the best and smartest thing to do.
On the flip side, diplomacy needs to be pursued….. long after this episode. Building relationships with our neighbors is one of the best ways to establish peace within a community and develop mutual respects. They too, have the power to file a complaint with the police regarding a barking dog and unfortunately, the cops are more likely to catch your lab in a barking episode than catching that kid with his BB gun.
You’ve got a delicate situation and if you want cooperation from your neighbors, you’re going to have to befriend them. If you can achieve becoming more than just some guy next door, you’ll get better results when respects aren’t honored.
Again, I’m on your side zoomer……. I wouldn’t feel any different or have any lesser a concern. From the details I’ve read, I just feel that the pathway to mutual resolve is a delicate one. Some people don’t think much of people, let alone pets.
Talk to the law and see what they advise. Beyond that, be patient, look for ways to help your neighbor, and be kind. Kindness burns like a hot coal on the conscience of a guilty offender. 
Good luck and I do hope you find your way to feeling safe again. 
