The neighbors came over to ask about some scrap wood for a planter, and Sage found an inner tube we'd cut up for some project, and that reminded me about the valve. So we launched a water bottle, knocked on the neighbor to apologize for the intrusion and retrieved our bottle, but then we got thinking about launchers, rather than doing it manually.
So Sage and I went into the shop to see what we could come up with. He learned how to use a drill press, and a Japanese pull saw, and we ended up with this:
Yeah, there's an additional piece of wood that probably isn't strictly necessary, but when a third grader tells you "I think we need a piece there", well, who am I to argue?