Most efficient way to achieve 3.6V or less from NiMH or Li-Ion batteries (when fully charged)
Good solution: 

2 x AA NimH AA from reputable maker (NOT 3).

Better still use low self discharge NimH AA.



2 mA x 24 hours x 30 days =~ 1500 mA hours.
  A modern good quality 2500 mAh AA NimH cell has a genuine capacity approaching 2500 mAh (say 2400 mAh to be safe).  In fact the capacity will be higher at the very low discharge rate of 2 mA.
 A Sanyo Eneloop AA has about 2000 mAh capacity.

To supply >= 1500 mAh the 2400 mAh call would need to lose no more than (2400-1500)/2400 =~37% capacity. Any reputable NimH cell will have less than that much loss in one month.
 An Eneloop or any other reputable low self discharge NimH call will lose far less than that in a month.

As you can tolerate a 1.9V supply voltage, using 2 x NimH makes sense. At 2 mA an AA Nimh will supply 1.2V+ over the very large part of it's discharge cycle.

With two cells and 3.6V max allowed voltage you can have up to 1.8V/cell - which you will never reach. Max ever is say 1.5V/cell on charge abs max, 1.35 V/cell  just off charge dropping rapidly to under 1.3V/ cell.

Discharge should not go below say 1.05 - 1.1V/cell at very low discharge rates. (Lower at higher rates).