@Johnex could you give us some details regarding what kind of PMIC is being used in the Mudi? That might be a bit more helpful for customised control (I have a gut feeling it communicates with the SoC over I2C, but without the precise manufacturer and model number, I can’t say for sure).
It should not happen - modern PMIC charge to ~80% capacity, then trickle charge until fully charged, and the unit should be running entirely from the external power source. Once the battery charge drops below 80% again (which, given it’s not being used, should not happen for weeks!), it will trickle charge again till fully charged, and repeat this process. Older PMIC, however, supplied continuous charge to the battery to keep it topped out, which was the source of explosions (PMIC receiving sudden jolt of power, providing overvoltage to an already topped out battery, which on the long run degrades the chemical reaction and leads to a thermal runaway).
Unless you somehow manage to discharge the battery while charging (i.e. making the unit use more power than what the external power supply can provide) multiple times, it’s doubtful that it would happen. Also, given the above internal view, the Li-Po cell should be straightforward to replace (it’s a dual parallel cell LiPo with the standard 5 pin connector), as long as you can find a fitting size.
Most likely not GPIO, but I2C - the PMIC already communicates a lot of data towards the SoC (battery status - voltage, degradation, discharge/charge status, and so on, external power status and voltage/amperage, et cetera) so that it can display that to the user. This communication might even happen through a serial port, but most definitely not just over a simple GPIO pin.
What happened after you pushed the other button? I presume it did not cause any nuclear explosions, and your cat is alive as well (if you have one), so it must be relatively harmless.