I've got a similar problem in the sense that the DFRobotUPS isn't powering up the Pi when the charge level reaches 7% — I'm powering it from an internal 5V power supply in an old Mac SE/30.  It charges up and a process on the Pi monitors the charge level and halts the system when it drops to 7%.

When I reapply the power, however, the charge level increases (I can see the level LEDs lighting up) but the power isn't restored and I have to hold the button to get it to return.  This is awkward as it's inside the Mac and I can't get to the button!

According to the I2C query, I've got firmware version 1.3, so I'm not clear why it's not automatically resuming.  Very annoying!