Protecting a circuit from the effects of a capacitor's charging
My money is the fact that the "batteries" are in parallel.

In actual fact, what you have there aren't just batteries, but batteries with boost converters.

If you open them up you will find a LiPo battery at (around) 3.7V. That feeds a boost converter to raise the voltage to 5V.

Putting voltage regulators (of any sort) in parallel is never a good idea as they will basically fight each other for control over the target voltage, resulting in nasty things happening.

You would be better off splitting your system into separate power zones with different groups of LEDs powered off different batteries.  That way they only share a common ground, not common power.