I received the new (second) battery today, and installed it.
Before I started, I charged each battery individually on the board, then after they were both hooked up,
I plugged the charger back in to verify that the charging circuit still worked. It took a while to get the pulsing
light, but it seems to work fine.
Here are some pix:
The two parallel battery sockets.
The stock battery is held on by some gooey double sided tape. It must be removed to provide
the room for the second battery, so I used a thin-bladed knife to cut it in half. It's very sticky,
so don't try to simply pry it off, or you could end up with a two-piece PCB.
I used my thumbnails to remove the tape from the battery, then reinstalled it.
I used some double-sided Scotch tape between the two batteries.
The board is ready to be reinstalled.
As you can see, there's barely enough room for the two together.
Back home.
The Mouser label with part number.