I think with this question you need to see the implications of what you are missing out on. As already mentioned the multimeter seems to be totally the wrong tool for the job and it will lead into a false sense of security. Seeing i believing in this case, show the results on an oscilloscope through a uCurrent and it will become clear where the multimeter has it's limitations.
Please enlighten me on what I am missing.
To explain what I don't understand, I must restate my point:
The same way one can adjust or measure burden voltage is the same whether one is using uCurrent or DMM. The OP can connect this two dots.
So, what I said without being too blunt was: use the dmm, dig around and try to measure the burden voltage of uCurrent. If it is too low to measure, the OP got the answer "does it matter" which was the question posted in the OP. In the process, the OP will figure out exactly what the best next step is.
I can't think of reasons to know exactly "this nano-second what the current draw is." So I assume in all likelihood, the average, or the min/max, or the trend are a lot more important for the OP. But he will have to get there himself since he knows what goal he is hunting for. Digging around will help him connect his dots.
Is just an Arduino. What risk is there at 1-2V 40uA? Perfect opportunity to dig around and learn.
So, with my point restated more bluntly. My question is: So, whats wrong with digging around a bit to answer one's own question? Exactly what am I missing proposing that? (No sarcasm intended) I sure like to know if I am missing something big.