First off, your drawing is a bit mislabeled: you have a BMS board, not a BMS chip alone (see your previous question); and what you label "VDD" is not what VDD should be, namely the positive supply voltage of a digital chip, but is probably "GND" on the MCU board.
Because we really had a bad time understanding each other on your previous question, please go through every single sentence that follows and check whether you understand it. If you don't, or have any other questions, ask about that first, before asking anything else.
Now to the multiple problems:
Ground shift due to abusing a battery cable contact
In a battery management system, large currents flow.
In your case, large current flows on the B- cable.
A large current means that even with small cable and resistance, there can be a potential difference: Ohm's law.
So, even if there is a cable or trace between the point where you connect and your BMS's ground potential, there can be a substantial potential between these two points.
So you connect the ground of your MCU board to something that is above the voltage of the ground of your BMS board.
That means that, for example, the voltage that the BMS pulls the SDA line to to signal a "0" will be below the ground voltage of the MCU. The MCU's protection diodes will conduct. If the potential difference is high, too much current flows from MCU to BMS, and your MCU burns.
Externally forced potential difference
The external supply attached to the MCU makes matters worse: If that is not isolated, it will pull ground of your MCU board to some specific potential.
Your charger, if not isolated, will set the negative charger rail to a potential, which might be the same as the GND from your "Power Supply" block.
Now, between that CH- contact on your BMS board and the B- contact, there's a current sense shunt: So, by design, the B- pin is above the potential of CH-. If CH- and power supply GND are connected externally through power supply and charger supply, then the same current as through the current sense resistor can flow, one way or another, from MCU to BMS, and something burns.
Solutions
Clearly, don't use B-. It's the least sensible connector for ground in your system! Simply don't.
Then, instead of using an external power supply for your MCU, simply use the voltage supplied by the MCU for controller things. That's what it's for! If that seems impossible, use an external supply that you know is well-isolated, or simply use a separate, small battery, to power the MCU.
