Take care poking about inside, lot of electronics in there sensitive to static, so earth yourself on something metallic (e.g. radiator) before you touch the bare board.
Inside you'll see the small battery wired to the board with two small flyleads, which are colour coded red for positive and black for negative, snip these one at a time, and then solder on some extensions, so the lead can go outside the case. Heatshrink the connections and solder on a small plug to the flyleads that matches what you have on the external battery. Then you can plug in a single Lipo of your choice. The protection circuit on the battery can be recycled if you wish, or you can simply time how long the camera has been running, and turn it off before you run the battery too low.
I would remove the old battery if you're going for a larger battery.