Yuck, I'm so sorry you have to deal with that!
Can you tell him it's fine to do the camping trip but not in the backyard as it damages the grass-- so please go to a campground? Seriously, it's like 25 bucks a night!

Well, letting the neighbors and/or the police know about it would pretty much take you out of having any responsibility for it. Even if you were there, he has clearly stated he is going to do this anyway. Short of pulling something more lethal than a BB gun on him (which I wouldn't recommend) I see no way you have the ability to prevent it.