In the Video settings, after disabling async compute and restarting, the GOG overlay started working for New Colossus. I already had deferred rendering disabled. As for the the rest of the video settings, I had vsync off and everything set to Ultra/High but I don't think those settings affect the overlay.

In my case, disabling async compute worked. Since its description mentioned rendering I thought "why not". Hopefully it's the same for others. For the sake of transparency, I have a mod to skip the intro movie and just head to the main menu after booting up, but that's for my sanity.