Alkaline AA discharge curve at just a few mA (to replace a CR2032)?
Typical Alkaline AA is 2000-2300mAh.  Self discharge is around 2-3% per year or 60mAh.   

I've tested 100 mA discharge rate to 1.1V, yielding around 1800mAh.  For so much lower than this, I would expect you get very close to nominal capacity.  This would be around 800 hours.  

This is assuming your mean load doesn't include high spikes.  Alkaline internal resistance is around 0.15 ohms.  Lithium coins cells are around 0.01-0.04 ohms.  So their losses for short spikes will be a little larger than the coin cell (and overall run time decreased slightly).  If the current draw is fairly constant, then this isn't a factor.

Ideal system would be using Eneloop NiMh cells for this.  They are generally around or below 1.2V under "normal" use with higher current.  With this light of a current, they will hover around 1.3V for most of the discharge and get around 1900mAh before dropping through 1.1V.  Self discharge over the month would drop this to an effective 1800mAh or so.  This should give you around 30 days of use and a recharge to get back going.  Best option for both the pocket book and environment.