If you ran it with one battery would you not have the problem ?
The problem shows up when you try to "hot-swap" just one of the batteries and replace it with one at a different state of charge, as the README claims you can do.
I think there is no shortage of battery management ICs, but the number that can arbitrate between external power/battery A/Battery B certainly eliminates 97% on your Digikey parametrics.
https://www.analog.com/media/en/technical-documentation/data...