Thanks for the suggestions. I put in a new Energizer battery and that wasn't it. The same behavior occurs in "emergency" manual mode, so I'm guessing it's not electrical. It appears to have the same pin mechanism that the RB has. Unfortunately I don't have another back. Is it likely to be the back since the back works if it's turned the other way? I ask because it appears to use the same linkage on the back either way.
I had the same problem on my RB and I always had to use the manual slider to trip the exposure indicator. The RZ doesn't have one of those so it's more of a pain.