Optional steps:

Due to the limited length of the leads coming from the PCB and the difficulty fitting all the wires in the casing, I decided to attach a connector to them to attach to an external battery. Any 3 lead connector will work. You will have to put a small hole in the casing of the battery pack.

Use the same steps to splice the wires to the battery to splice in a 3 lead connector instead. I recommend splicing in the female end of the connector and keeping the length of wire from the PCB and connector fairly short.

Splice together the wires for the replacement battery to the male connector. Make sure to leave enough wire length to connect to the battery pack.

I used a 1500mAh pouch battery to give myself a much longer working duration between charges.

Note: the colors of the wires from the PCB and battery may be different from what the wires from the connectors are. If you have to connect black from the PCB to white of the connector, make sure to connect black from the battery to white of the connector.