Found it.

This is the electrical box in the shower. The black wire circled (yes, that's my attempt at a circle and an arrow) is shorted to ground somewhere. You can see it goes into the ceiling framework and vanishes. I figured that logically the next stop in the circuit is the light in the shower. It is. I took the light out and disconnected the wiring. No more short. So the short is between the box and the first light in the circuit. I ran my own red wire (arrow) to the shower light. I now have all lights and fans working!
I got lucky, sort of, in that the existing red wire runs through an accessible conduit right by the light. I can run my new wire in there to bypass the shorted wire.
It's still disconcerting that that wire shorted out on its own somewhere. I would have been happier if it was a mistake I had made in wiring the fans.
And the moral of the story is don't always trust the labels on the fuse box. It was NOT a spare fuse, and the circuit marked "LIGHTS" isn't.