Hey toothsaw, when you plug in the battery pack you do several things. You got from wall AC/DC adapter straight to battery DC current. You also change the amperage. The value of the adapter is well explained here, and if ever there is a reason to prefer Apple original vs fakes is here: http://www.righto.com/2012/05/apple-iphone-charger-teardown-quality.html?m=1 . The 6 can only do 1.6 Amps, the wall cube (hopefully original) 1 Amp. Your battery case, something else- likely within specific specs, you might know that but without hooking it up to measurement tools hard to tell. I once had to buy a convenience store charger as I misplaced mine, after I was done I threw it away.
Power rating for iPhone 6s charger ?
http://www.apple.com/ca/shop/question/answers/iphone/will-a-21-amp-charger-damag e-my-i-phone-6-s-plus/Q2DXC4PTJ7KADKF9Y
What all of this means is that if you get different results with different currents, the correct lab/class troubleshooting solution is to suspect immediately the battery. I do understand that it started with the iOS- itself with own amperage, but the technical step is always the same. Apple Support at Genius Bar or Sr Apple Support should be able to assist you and your issue will be resolved. Perhaps you are over 600 cycles, and it all comes down to the same technical steps.
A bit of anecdotal evidence, had a new iPhone 4 dying in a month a few years ago. In my case I had bought with the iPhone 4 a Belkin car adapter from the Apple Store and was doing hundreds of miles of Hwy driving that month, phone plugged in. I lost 3 car fuses in that month, and the iPhone right after. Fuses # 2 and 3 were my experiment to isolate the issue- it was the Belkin Adapter, one fuse per long ride; provided Belkin feedback, Apple reimbursed me, shortly after they were gone from the store. Could not have been isolated. Since then, minus that convenience store emergency, never again let a non-Apple device providing power to my Apple device; and I cycle the Apple adapters once in a while.