At JFK-
Since the Soho and Greenwich lounge receptions are separate, that gives you two sets of agents to ask nicely if they'd let child #3 slide in.
The Flagship lounge offered a day pass but I don't think it has been clear if Greenwich does? Admirals Club would offer a day pass but debatable if that's really worth it if this is late afternoon/early evening during the Europe redeye peak as it gets pretty crowded in the small space.