The battery is a 3.6Whr Li-Ion one and the Sony Part No. is A-1832-983-A; as far as I know it is not a standard form factor. It should be available from Sony, and maybe from other places, but I have not had any need to look: someone else may have a lead, but otherwise having the part number should give a lead though if not going to Sony.
In case not known how to get into the device (and no responsibility taken by me in the following
) there are a lot of internal clips inside each edge. With the reader face down on a cloth or similar on a flat hard surface use a unclipping tool (or make one from a thin piece of rigid plastic) and start at one top corner and work along the top, then one side starting from the top, other side working from the top, then the bottom.
As you say, 3 wires to solder to board.