Although it's easy to say the power source should provide 500mA, there's more to consider for the fact a battery is present. Its changing voltage affects everything. Its voltage opposes voltage from the charger. As the battery charges its voltage changes over time. And its ability to store electricity in the form of electrochemical energy changes.
The net effect can be calculated on paper... However the reality is you can't be sure what is the overall schedule of battery voltage versus charging current over time.
This is where you must plan to spend several evenings experimenting. Does a battery draw too much charging current at any time? Enough to be hazardous? Should you install a resistor in the current path? What ohm value? A transistor? 500mA is sufficient to make a small cell hot. You really ought to position a thermistor touching every Li-ion cell, to tell you whether a cell is or isn't hazardous.
These replies are coming from a distance, and I can't design an ideal charger for your battery pack.