Well, that was a disaster. The AirPort Express was apparently set to channel 6 and was set to something called "802.11a/n - 802.11 b/g (Automatic)" There was a couple of other options, one of which was the same but without the (Automatic) in the name. When I tried to change to that one, the Express stopped accepting my password, and then AirPort Utility stopped seeing the Express at all.
The Final Solution: Remove the Express 802.11a/b/g/n altogether and reinstate the Extreme 802.11 b/g "Mother Ship" router. I guess the Express is now bricked.