Hello,
I do think you are running into the common ground problem. The BatShare will not solve this as it also has a common ground inside it. There is really no way around this other than
charging one battery at a time or taking at least one battery out of the system and charging it separately.
The BatShare has two independent diodes in it, one for each battery. A diode will fail as an open (I have not ever seen it happen on a BatShare but that is how they fail) so the other diode will be unaffected and continue to supply voltage to the system.
Let us know if we can be of further assistance.