Doing some reading, it looks like CHIP just had a very similar issue a little while ago. The alpha version could idle or do a small amount of work at around 80mA, but the newer versions are >200mA.
We've started using raspberry pis as they will work on 80-100mA if you turn the ethernet and gpu off.
Ideally I'd be using a microcontroller as I'm not actually using the CPU resources (although having heaps of memory is really nice), but my country is about to have no 2G network, and all of the 3G options I have found that do not talk over USB add at least $100 to the unit price.