For bonus points do you know why my charger charges them to 4.2V per cell when at the 3.7V setting?
Adafruit has a nice article describing LiPo voltage range.
But why on earth are you putting resistors in series with the battery?
I tried putting increasingly small resistors in series with the battery but it only got the power supply up to 2.5V (with a 10 Ohm resistor which got very hot in a couple of seconds) so there's definitely something else going on here.
So let's be clear. If you measure the voltage directly across the battery it measures about 8V until you close the switch, then it drops to 1.5V? And when you measure the current what value of current does it show?
Try disconnecting all the components and adding them back one at a time until you see what's causing the voltage to drop. That's one way to find which is causing the problem.
It didn't go above 1 Amp and the battery should be able to manage 4A right? (2 cells x 2000mAh each x 1C safe discharge rate = 4A safe current draw?)