Parallel charge a serial pack, LiFePO4
As mentioned in your comments your drawing is not accurate so lets put that aside for now.

Yes it is theoretically possible to charge the individual cells in series.  To achieve this your will need 16 individual Li-ion/LiPo charges that are fully isolated.  

The output of each charger would need to be directly connected to each individual cell.  Then they can charge each cell individually with no issue.

Alternatively dismantle the battery pack and place the individual cells into individual chargers.  Thus allowing all the cells to be charged in parallel.

Advantages of the above method:  Fast re-charge times.  Uniform recharge times.

Disadvantages:  Requires extensive modification of the battery packs.  Either through dismantling of bring out wires from each node and the ends.  This does have risk.  What happens when the wires from the cell's are accidentally shorted?  As the wires can carry the full load and have a low impedance path to the cell's they pose a fire hazard.  So you will need to be careful fitting these wires and provide protection to them all.

Alternative is to charge all the cells in series with short circuit protection and only have the two wires coming out from the pack and have discharge resistors across individual cells that can be switched on for charge balancing ... also known as the BMS which you already have.

I strongly advise against making modifications to the battery pack to without seriously considering the hazards and mitigating against them.