Mystery solved! I posted my conundrum to Facebook, and sure enough one of my dorky friends came through. It turns out that I was very close with my guess: it's from Final Fantasy IV/2!

It's titled "Dancing Calcobrena"

I believe it's music that you would hear toward the end of the game, when your team is on the moon. I really loved that game but I haven't played it through to the end in about a decade (I've started it a dozen times one various emulators).

I'm still annoyed that not a single person involved with that episode of This American Life would respond to me about that. My guess: they didn't pay for the rights to this particular piece