Li-Ion Protection IC Always in Overcurrent Detect Mode
The AP9211 is described as "specially designed for 1-cell Li+ rechargeable battery pack application". This suggests that it is designed to be permanently attached to the battery, and may not be suitable for your 'unconventional' application.

But why does it not turn on if a load is present when the battery is connected? Regarding over-current, the datasheets says:-


  In discharge overcurrent or short current status, R VMS is connected
  to V SS but R VMD is not connected. The voltage of VM pin is almost
  equal to VDD as long as the load is connected. When the load is
  disconnected, the voltage of VM pin will become almost equal to V SS
  (due to R VMS being connected) and then the AP9211 will release from
  discharge overcurrent or short current status.


I suspect the IC powers up like this, with the discharge MOSFET turned off so even a small load will hold it in over-current protection mode. To use it in your device you may need to provide a power switch to disconnect the load when the battery is inserted (perhaps a 'soft' switch activated by very low current would do the job). 

If this is not acceptable then you might need to use a different protection IC which does not require the load to be disconnected at power on. However I looked at a few other protection ICs and all they appeared to have the same limitation. If the battery has to be removed from your device for charging then you might consider making a circuit that just protects against over-discharge (which could be set to a higher voltage for better cell longevity) and over-current protection via eg. a polyfuse.