OSEPP claims a 3-5v voltage range for that board, but the LM386 they have in the schematic is usually a 5 volt minimum part. I would ask OSEPP to confirm it works at 3V- it's certainly possible if they have some drop-in replacement for the LM386.
You should not need a bigger(capacity) battery for it to work, they specify a 4mA operating current.