Correct, when measuring voltage the current will be 10-15mA at most, however when balancing then the voltage drop will affect the whole module not just taking voltage readings.
So the voltage on the ATTINY chip will be bouncing up and down hundreds of times per second (because its driving a PWM output to the MOSFET). We already know that the ATTINY doesn’t regulate its internal clock very well when exposed to voltage changes, so that in turn will affect the communications to other modules.
Additionally, the voltage drop may push the ATTINY into brown out detection, which causes the ATTINY to reboot.