That is strange. I don't have an answer but I'll put out some feelers.
What is the maximum output of the fast charger? It should be written on the unit itself, like output 5v 2.1 amps, which would be 10.2 watts.
The mAH capacity of the battery should be available, and then one can estimate how much juice it will take to charge the phone.
BUt charging it while using it of course uses more juice.
Consider this measurement tool or similar, as it will show actual voltage
(USB sources work from 4.75 to 5.25 volts)
It will also show amperage and amp hours, well milli amp hours, at 5 volts.
https://www.amazon.com/PowerJive-Vo...UTF8&qid=1487184709&sr=8-7&keywords=usb+meter
There are many similar products, but some do not count mAh which make them less desirable.
I have a different clone of the above product.
Here is an In depth review on it:
http://lygte-info.dk/review/USBmeter KCX-017 UK.html
And I am sure you are aware of this, but you do not need to run the inverter just to use a plug in USB source. Many capable units fit into a ciggy plug receptacle, and while I despise ciggy receptacles, at 10.2 watts they will present no issues
My favorite 12vDC powered USB source is this:
https://www.amazon.com/Blue-Sea-Sys...&qid=1487185181&sr=8-1&keywords=blue+seas+USB
I use the older 2.1 amp version. The 4.2 amp version is much newer has a much smaller parasitic draw.
It is my favorite source as it maintains 5.12 volts under a 1 amp load, where my other sources struggle to hold 4.95v. This higher voltage in combination with a shorter quality micro USB cable charges my S4 mini battery much faster than my other sources, but I have no hungrier USB devices that could load it at 2.1 amps.
The 12vciggy plug/powerport USB styles are numerous too.
https://www.amazon.com/s/ref=nb_sb_...eywords=12v+usb&rh=n:15684181,k:12v+usb&ajr=0
One thing which I have noticed also speeds charging is a clean micro USB connector.
Consider acquiring this kit, as it is so useful when any electrical connection comes into question.
https://www.amazon.com/DeoxITKit-In...1&sr=1-1-catcorr&keywords=caig+industrial+kit
These Caig DeOxit products also lubricate and reduce wear on the micro USB connector, which appears to be the downfall of many a smartphone whose owner does not feel the need to upgrade with every new model.