one man said,capacitor is more powerful than dynamite
I agree with your idea that the batteries must do two things it seems logical -but as far as I am familiar with capacitors They have suction power or something like a vacuum,because of this the effect when we filling large bank of capacitors or individual capacitor we have to have some kind of resistance which corresponds to our power supply otherwise we would destroy power supply or source.so capacitor has interesting properties and I believe that the battery does not have to do two things.
Another example perhaps simpler than capacitor but in general is same.
we have a charged battery 12V , 100Ah when we take another same battery but completely empty,If we connect the two batteries in parallel it will happen 50% of capacity equalization ,before that happens we can connect some kind of load between terminals + + or - - to make things even more interesting it can be some switching transformer that can charge a third battery,and it has to work to the point of equalization of capacities and because of the potential differences.at the end of the process we are left with the initial capacity(two batteries with 50% capacity, which can be for later use connected in series) +some amount of energy in the third battery.
now it is more than crazy and I really do not want to make any problems here(on forum) I just drain all standard ideas how to solve this problem and really looking for someone with a higher education for help,and this forum it seemed like a good place.