Good news! After posting this thread I went out and got a rivet to stick in that spot. Before pulling the rivet I bummed firmly on the rivet gun a few times to get it good and tight and it had a good comfortable amount sticking out. My son held the acorn nut tight as well. I will get around to replacing all of those over the fridge
and firm them up as it will surely happen again if the others are a little loose.
I was actually fully prepared to replace it with a 10-24 flat head machine screw, a neoprene washer up on top for angle cushion, a 5 mm lock washer, and an acorn nut that was not much smaller than a 1/4-20. I will keep that hardware around for backup or the give up if it keeps falling out.