Batteries Spike in Watts and Amps Pulled with less than 50% Life
When you are performing test and/or other research activities, it is customary to qualify and calibrate all your instruments with known good sources and loads. Please do this with your power meter. 

Get/rent a good benchtop power supply with logging capabilities (to record voltage and current), and get a known good load, either electronic load or a good resistor, and take readings of your power meter over a good range of loads and supply voltages, and check if they match the external (calibrated) numbers. 

Only after that you may start researching behavior of highly non-linear sources as Li-Ion cells and highly non-linear loads as LEDs. For research of time-variable sources and loads, it is highly advisable to have meters with logging capabilities, so you will be able to correctly get integrals over time.