Thanks everyone for the suggestions. I admit I skimmed that sparkfun article a little too quickly. I'll go back and read that more closely. It sounds like it might be good enough so I'll look into it some more.
You said, that you want to automate stuff. I assumed it is for work. When the employer wants something reliable for production, it is not that expensive. Anyway, for home use, I would not buy one of course.
The LTC4150 is not really suited for this, at low currents it will have the same issues as any other system.
This is for my employer, but my employer is me
I used to work at a big company that did buy tools that cost over a $1M (one example being arrays of FPGAs that simulated future chip designs pre-tapeout). Back then I had a very similar mindset where if a tool cost $10K and it would help an engineer at his job it was an easy decision to get it. That said when I'm going small productions I have a much higher bar for a $10K tool. It has made me appreciate less expensive tools that still get the job done. The smarter and cheaper tools are a big reasons kickstarter like projects can exist today. If embedded programming still required people to get a $10K compiler a lot of cool new projects wouldn't exist.
To me it's sounding like the exact tool I want doesn't exist, but I'll check out that sparkfun tool more. It wouldn't be that hard to take one of those inexpensive 2 channel USB scopes and combine it with something like Dave's microcurrent. Then write software to integrate the total power consumption by collecting something like 100K samples per second. Sounds like a fun project if I ever have time.