Bipolar Li-ion battery charger
Theoretically you could use a two cell charger to charge both batteries in series. However If your device is not drawing identical currents from both batteries then they may become seriously out of balance during use, so the only safe option is to use two 1 cell chargers that charge each battery separately.  

If both chargers shared a common ground then that would have to be connected to the battery center tap, and one charger would have to put out negative voltage and current. 

Alternatively you could use two identical chargers, but with each one having its own isolated supply so their 'grounds' don't short out the lower cell. 

Dual isolated DC supplies can easily be created using a mains transformer which has two output windings. You simply attach a rectifier and filter capacitor to each winding. Here's one that I designed and built many years ago:-