So good news! Someone on another forum mentioned that the internal button cell battery might need to be 3V, not 1.5V, and after some digging around the web I found that to be true. I de-soldered the battery and soldered on a couple leads that I could use for testing by applying 3V with another battery. I assembled the unit and installed the AA batteries, but before applying 3V to the leads I tried to turn the unit on. Surprisingly the unit turned on! But I am really puzzled how this is possible without the 3V button cell installed, as everything I read said without it the unit wouldn't turn on. There is a reason the button cell is there so I'm definitely replacing it, but I'm curious why the thing turned on without it.

I also measured the button cell again after removing it and it was around 0.5V. I'm not sure how I got 1.5V before, but I'm guessing it was a residual charge since I initially measured it shortly after removing the AA batteries.

The button cell only had markings "68 f+" which didn't seem to help me. I found a page about replacing this battery and the person mentions part number ML621-TZ1-D, so I'll probably just go with that. It's a 3V lithium cell and the only difference may come from the capacity. The ML621-TZ1-D has a 5.8 mAh capacity. [Only registered and activated users can see links. ]