When developing my UPS I had your exact usage scenario in mind.. among a dozen other things as well.
Run the numbers and see what you can expect.
5 minutes per day at 10 watts (this is on the high side for a Pi).. gives you 0.83 watts used per day
Idle power.. the UPS itself takes about 1ma when powered off.. 0.17 watt used per day
Your total power usage is around 1 watt per day.
NIMH AA batteries assuming 2.5AH, 6 of them of them will give you 18 watts total. Gives you 18 days on six AA batteries.
C cells usually double the power (5AH) and D cells usually double again (10Ah) so the math is pretty easy there. 36 days on C cells or 72 days on D cells.
You can use a common 12v 7 amp hour lead acid battery too.. that has 84 watts or 84 days of power.
There is self discharge and other things to consider if you want to be picky.. I would just subtract 5 or 10%.
Here is my UPS, check the PDF manual, specifically the "UPS Timer Control Settings".
You can also use a small solar panel (1-2 watts) to keep rechargeable batteries topped up. Since your power usage is so low it will likely run a long long time even in sub optimal charge conditions.