Based on my experience with the uCurrent, both the input and output -ve terminals are connected to ground (referenced to the power supply). I didn't realize this and couldn't understand why what was a perfectly functioning system without the uCurrent turned into a system that shorted the voltage I was supplying to my DUT. In my situation, I used an external supply rather than the battery. So, I'd expect Method 1 to fail. I don't know about Method since you have the uCurrent on the low side of the DUT. However, you shouldn't assume that the voltage waveform and current waveforms would be the same anyway. You may have capacitive/inductive effects going on with your DUT that would certainly affect the appearance of the waveforms. Without seeing example 'scope traces, it's impossible to say. Another point - are you actually measuring your DUT output voltage or are you assuming what it should be?
(My memory is a bit sketchy on this since I managed to overcome the problem and move on with my project
The uCurrent is a great device)