Battery Management Controller
Hard to say.
UVLO is not the same thing as overdischarge protection. From the datasheet of the controller you provided, UVLO monitor the difference between the supply voltage and battery voltage. "The UVLO circuit places the device in Shutdown mode if the input supply falls to within +50 mV of the battery voltage. Again,
150 mV above the battery voltage the MCP73831/2 become operational".


The UVLO kicks in when external supply falls to near or below battery voltage. The controller you mention is a charge controller, and has no function during battery discharge. If your cell voltage drops to 2.6, you plug it in to charge, MCP73831/2 checks the supplied voltage and it's 3.5v or more, it starts charging the battery. If the supply to MCP73831/2 is not good enough to keep up with the current demand when charging and the supply voltage drops, the controller stops charging. Check you charger supply voltage and power ratings.