What is the peak current for CR, LIR and AA batteries?
Alkaline or NiMH AA batteries are certainly OK at 121 mA. The current draw in a speedlight/photoflash is higher than this based on the big cap charge time. 

4 AA batteries can charge a 1000 uF, 300V cap in about 5 sec -> 0.5*C*V^2 = 45 Joules = 6V * 1.5A * 5 sec if my math is correct.

121 mA is more than I would draw from a coin cell. This Energizer "pulse" test only draws 7 mA. http://data.energizer.com/pdfs/cr2032.pdf  Even if they aren't damaged at 121 mA, the internal resistance is likely to be an issue.

Edit: OK, here is some data on the autonomy. An Eneloop NiMH rated at 1900 mAh was tested to have a capacity of 1877 mAh at 200 mA. Therefore, there is minimal loss in capacity at 200 mA, slightly more loss at higher currents.  https://lygte-info.dk/review/batteries2012/Eneloop%20AA%20HR-3UTGB%201900mAh%20(White)%20UK.html

Edit2: Discharge Time Calculation:



It is easiest to determine how many 6 hour cycles it will last. The calculation isn't hard, but you need to be careful with your units. Here everything is in hours and mA. If I made a mistake, the eagle-eyed people here should catch it. 

Note that batteries will discharge even with no load. Modern batteries are quite good, they will go for several years with minimal loss. Still, I would shoot for 2X margin in your situation (calculation should show > 2 years).

If physical space is at a premium, consider AAA batteries.