You can't charge at 6V and draw power at 12V at the same time. You can wire the batteries in parallel for 6V or series for 12, but not at the same time.
The only way to make this work would be to put the batteries in parallel to charge, then disconnect from the mill, then put them in series to run the inverter. That could be done with big switches, but they would need some interlocks or custom design to avoid shorting out the batteries. This might be practical in special situations. An example close to my heart would be charging the battery (for radio and instruments) on a sailplane all week, then flying it on the weekend.
If you need to draw power while charging, you could use a boost mode switching regulator to increase the 6V mill output to ~14V for charging. Look up the LT1070 data sheet/application note for lots of examples.
