Well I think I figured it out now: The 2 headphones are connected in series (white cable) and the mic is connected separately (black cable). Both black and white cable are connected to the headphone input jack. I call the headphone piece in your image the left piece. The connectors on the left headphone piece I call 1 to 3 from left to right: 1.) Left headphone V+ 2.) GND 3.) Right headphone V+ (it's just soldered there to hold it in place, it's not actually connected to left headphone) The wiring to the jack is exactly as on the photograph of @jayeff
I did exactly as you described. My Shield booted with the new battery and charged to about 50% - then it just died. The 2 power lines going from the battery board to the shield show exactly 0V - no idea what went wrong. The battery seems fine with 3.8V. Can you help?