No that doesn't work. If you want to charge the top cell switching Q1 on (without current limit!), your Q2 (also no current limit) will pull Q1's base below the cell's negative voltage.
This is your circuit for one cell. The signal coming from the microcontroller switches on Q2, but also is pulled down to 0.7 V. So the base of Q1 is also at 0.7 V, while it needs at least 5.1 V (0.2 V saturation voltage of Q2 + 4.2 V battery + 0.7 V base-emitter junction).
A solution could be to have a current output boost converter. This can generate a higher voltage from a lower input voltage. Constant current boost converters are often used as LED drivers.
Replace the LEDs with your battery, and set the charge current by choosing R1 = 95 mV/I.
There are very expensive $250- $300 chargers that do higher than 6s, probably closer to 10s, but not a lot to choose from and right now my memory is drawing a blank on their names, but they do exist.
I have no idea why there are not more, but I suspect that the demand is simply not there yet. Lithum batteries at those higher voltages are not as common and can be very expensive as are they chargers.
The charger you link to balances its battery by having a charging connection and balancing connection at the same time. The charging connection is directly connected to the + and - of the battery and supplies the main charge. The other connections are more complicated and, for example, in a 6s battery there would be 7 connections, one at the "-", on at the "+", and a connection or wire coming from every single cell connection. So each time another cell is added to make it a 2s or 3s, a wire comes out between the "+" and "-" of each cell added. So a connection between all 6 cells and one at the botom or "-" and one at the top or "+" and you have 7 wires coming out that will then plug into the side of the charger.
The charger then monitors each individual cell's voltage as it is charging the battery as a whole, but most chargers don't seem to start balancing until the battery is essentially full, or at least one cell is at 4.2 volts. Then it uses the seven wire connection to balance the battery, usually by discharging the higher voltage cells a little via a small current, and then charging the whole battery again slowly. Then repeat until all balanced.
It looks like what you linked to would work, except that they are for smaller number of cells in series than what you want to do.
Another option that would work for you is to do what you are doing - split the 10s into 2 5s and charge them independently, but parallel charge them using a parallel charging board and then you could charge them at the same time.
Check this out: http://www.hobbyking.com/hobbyking/store/uh_viewItem.asp?idProduct=14856
Best Answer
While 'charging while using' is possible for a LiPo battery, it is not recommended, as it may compromise the battery safety and lifetime.
This is because the charger measures its own output current, but it does not know about the load current. The battery voltage is also lower than it would otherwise be. This means the charger incorrectly estimates both its state and rate of charge.
The idea of a standard charger charging in parallel is overwhelmingly unlikely, and can be dismissed.