Not sure what sketch you use on them, but I would check that the battery is around 3.4V to consider it almost drained, for the purpose of shutting off your Pi.
For a typical setup even a 2000mAh should last a pretty long time, maybe 1-2 hours.
Chinese batteries can sometimes be flaky, in my own experience. Extreme temps (in my case below 32F) will affect them, I can only assume extreme heat will also harm them.
So how they respond under a dynamic load can influence their performance long term.
If you can, hook up an oscilloscope to the battery volts and see what happens if you pull the plug. They should hold steady as they slowly discharge if they are in good condition. If there's a mess of noise or spikes and drops then the battery is no good, and the battery checking code can read one of those drops and issue the shutdown.
The MHat uses the MCP73831 charger, it's a simple setup and default is a light 212mA charge current.
I would probably not recommend using batteries in parallel since the charger was not designed to charge parallel cells, only 1 cell. I would first try another battery if you're having issues with your current one.
