Three ways to deal with this spring to mind.
1: Use a switched coil voltage doubler (like a switched mode PSU)
2: Use the "charge individual capacitors in series" approach
3: Charge individual batteries in your chain approach.
1: Is probably the most efficient, will give you the least voltage issues but is complicated to home brew.
2: Runs the risk of voltage dip as you switch between capacitors, switch too fast and you waste time connected to nothing or run the risk of shorting out a capacitor
3: Simplest but runs the risk of not changing each battery in your chain evenly (this would be reduce the life expectancy of the batteries)
I just drew a drawing of the curcit you need to do 3 but "insert image" turns out not be insert image at all but "insert link to image hosted somewhere else"