If AA batteries don't supply sufficient power, will adding more of them in parallel help?
At 1.6A you're shorting the battery. Maximum load is about 500mA, this is already &lt;1 hour.
Internal resistance is about 0.3 Ohm, at 1.6A sets you back half a volt per cell. So, for one pack with 4.5V (real 4.2V ) you go down to 2.7V in real conditions.
If equally divided over 4 packs this 1.6A is maybe fine (0.4A per pack), but if one cell fails, your pack cascades into failure. In theory with 4 packs you have 0.1 Ohm internal resistance, which is about 120mV drop during you worst case condition. For the entire pack that is about a drop to 3.72V. (4.2 - (0.12*4) when full. In theory.
Can you retry and measure the current of each pack vs the supplied voltage? This should give you insight in the paralleling behavior.

I also suggest rethinking this backup solution. Either

Put all the 12 cells in series and use a buck converter. (less current, same power, better power point)
Use C or D-cells.
Use protected lithium cells eg: 14500 (in series).
Maybe even consider lead acid if that fits.

Now you're abusing the poor AA's.