Dual power supply for low power circuit
Why don’t you put the batteries in series, use a buck converter, aim for the lowest system voltage to squeeze out max. Battery life, so aim for 2V5?

This way you have no cross charging and your current problem with your LEDs doesn’t exist anymore as it’s constant voltage.

Also 2V5 seems like a high-ish system voltage for BLE and MCUs, are you sure you can’t go lower? LEDs probably are the driver here.