I found a battery.
https://www.ebay.co.uk/itm/3-7V-500-mAh-3-wires-thermistor-Li-Polymer-Li-ion-battery-for-GPS-Camera-902030/183606690932
It’s the wrong shape, but I made a small hole for the wires in the side of the battery compartment, and I hotglued it just outside the compartment.
Unfortunately the battery light still stays on even when the headset is off.
I think there must be something in the firmware which turns on when a battery reaches end of life, and I don’t know how to reset that despite installing the new battery.
However, the headset turns on now and seems to charge correctly.
I’m going to see if I can reset the firmware somehow.