Remove over-current protection of battery protection circuit
The circuit looks for voltage drop across two power mosfets. It looks like the sensing circuit is between CS pin and GND pin. 

Instead of looking for MOSFETs with lower Rdon, you can make a divider, splitting the R2 in two resistors, and connect the CS pin to the middle of the divider. The voltage between CS and GND will drop, IC will think that the current is small, and will trigger the overcurrent threshold at higher drop-through voltage. 

Just try to connect another 1k resistor between CS and GND, and see if the cell will deliver more current before cut-off.