This turned out to be a step down the correct path, even if pretty embarrassing in the moment. Someone with a better relationship with the kid stepped in and used that relationship to de-escalate, á la "please don't do that because it will hurt you and I wouldn't like that because I care about you."
What I found to be key in a situation like this was: (1) focus on the person, not the behavior; (2) Have full attention on the situation; (3) try to determine/address root causes. It's obviously impossible to do all that stuff in real life sometimes, but I can try.