Writing back to say that Xpadder worked. (Thanks for the recommendation!) It took a minute to learn, but easily handled the task of mapping Polycade's left stick to a D-pad. The tool should come in handy for other games too.
One limitation is that it won't work with two player games because the controller maps onto the keyboard, and there's only one keyboard. Perhaps others are crafty enough to find a workaround. To play Unravel Two, I ended up getting an Xbox controller for one of the players, and then re-mapped one of the Polycade joysticks for the second player. I'm playing with a 3 year old, so he gets the Polycade station 🙂