Seems obvious that the module is designed to run from single cell lithium power.
4.2V -> 3.0V before the voltage begins freefall.  Also, you get better
life out of the rechargables if you don't let them go as low.

In the McGraw-Hill handbook, they show a 18650 cell curve on page
35.37 that seems just fine.. You'd need an efficient LDO, but you CAN
shut that off when not in use, so it's not always blowing away
microamps.

Use the pic to enable the power to the zigbee, then talk, then shut it
down again.
If you need to receive while "asleep", you can probably do that by
recieving only once in a while, and sleeping most of the time.


Non-rechargable Li cells are an even closer match, with Li/MnO2cells
being essentially flat at about 3.1V till they die.  14.59 shows pulse
discharges of 5.5mA into a 190mAH coin cell.

Sounds like they want to be fed from a 123A cell..
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