Thanks for all the replies. I hadn't thought about asking a random person coming in at the same time if they'll guest our third child if the agents don't let them in, great idea. I would certainly do that if I were solo and saw the reverse situation, no skin off my back.

Fortunately, we don't have too much time to kill this trip (taking the morning BA flight JFK-LHR, so would really only be stopping into the lounge for a few mins to grab some breakfast) and not too long of a connection at LHR on the way home, at least have the LHR Admirals Club to fall back on (I have a membership). Moreso wondering if we're going to be stuck with this issue every time we travel internationally for the next 10+ years because we decided to have three kids instead of two!