Using balancing charger with Li-Ion protected cells
Can your 2s charger not charge in normal or 'unbalanced' mode with the 3 wire balance connector not connected? Almost all chargers do.

So why balance charge your 2s battery? The TP4056 chips should ensure that they each charge to the correct 4.2 volts and therefore will be balanced.  Or will th TP4056 not work when charging in series?

Even if you connect the balance wires, it should work. All the balance wires do is measure the voltage and then at the end, or near the end of the charge, bleed small current from the higher voltage cell and then recharge the battery up to 8.4v.  So can you discharge a small current (usually less than 300mA) from the TP4056 connector? If so, then I believe that it should work.  If the TP4056 connector doesnt allow you to measure the cell voltage, then the charger won't even start.

However, I have no experience with the TP4056 so watch it closely if you decide to go ahead!