Electronic – How to modify USB battery pack to function like a UPS

I bought this USB battery hoping to use it as a uninterruptible power supply for a Raspberry Pi 2.

I researched it, and users said it could be charged while also powering devices…just what a UPS should do. However, when I tested it, I found a big problem. Even when the battery is fully charged, when I plug or unplug the input cable to the battery, there's a brief voltage drop on the output plugs that causes the Raspberry Pi to reboot.

Is there an easy way I can fix this? I've pried off the housing, and it's three of these cells wired in parallel onto a fairly large and complicated-looking control board.

Would soldering a large capacitor onto the VCC/Gnd pins of the battery's output plugs work? Or should I attach a capacitor the the RPi's power input pins? Or would I be wasting my time?

How would I measure the length of the voltage/current drop, so I'd know what size of capacitor to use? I tried measuring the DC signal with my DSO (DS202), but I could figure out how to detect the length of the drop.

Edit: Using an Arduino to measure Vcc with an ADC pin, I found unplugging it causes a 1-3 second brownout, while plugging it in causes a 1-3 millisecond brownout. I was able to fix the plugging-in brownout by placed a 4700uF capacitor across the Vcc/Gnd pins of the output USB jack. However, the unplugging-brownout time is too large to fix even with a supercapacitor. What other options do I have?

I also think I found a fault in the control circuit. When the pack is unplugged and running solely on the battery cells, the output voltage is 5V. However, when it's plugged in, even when fully charged, the output voltage is 4.7V.

Edit: It seems the plug-in-brownout was in part due to a bad external USB power supply and cable. I was using one of those cheap small cube wall adapters with USB jacks in them, with a long USB cable. It says it's rated for 5V@2.4A, but the voltage was measuring between 4.5-4.7V. When I replace it with a USB single-piece wallwart, the voltage measured 5V.

So now the only remaining problem is the 1-3 second brownout when the external power is removed, likely caused by the controller slowly switching circuits to the internal battery. How can I fix this?