I think the NPN has the collector/emitter reversed if you consider the battery a load for the charger. Using a MOSFET may be better. You also need a current limiting resistor on the base, and perhaps a base bleed resistor between base and ground, too.
What is supposed to happen when the switch is left vs right? As far as I can tell, when the switch is left, it will attempt to parallel the two batteries, whereas when it is right, it will attempt to series them.
Btw: I would not hook that circuit to the rest of the pieces until it worked exactly like I want. Too much risk of burning things!