the battery contacts only connect with the circuit board when the two halves of the housing are in contact, so there's no way to get inside... although I suppose there might be a way to grab the battery contacts with some sort of clips. The battery contacts are fairly flush with the surrounding plastic and don't leave much to grab.

I was looking at the flex cable and connector more closely yesterday, and I think/suspect that the connector is working. I can see the upper half of the contacts move down when the latch is in the closed position.
The other end of the flex terminates on the LCD contacts, which are just a near-transparent layer of metal. No idea what bonds the flex to those contacts.
It would be interesting to see if pressing the flex harder against the LCD contacts makes a difference, but that would require that the board be powered up.
Now that I think about it... maybe I could just power the board from a power supply instead of the battery??

Steve in Peoria