Based on the extremely low temperature you've described, it's likely the temperature is causing the drop-out of your system.
According to All About Lead Acid Batteries, the voltage you read at lower temperatures is not accurate. The voltage shown on the meter will by higher than reality. The chart shows that at zero degrees F, a battery that reads 12.082 volts is effectively fully discharged. Extrapolating from the chart, at minus twenty degrees F, your battery is dead even though it might display 12.14 volts. For high current draws, it is inadvisable to discharge a lead acid battery below fifty percent depth of discharge, as it results in reduced battery life. At lower current levels (your application) going below twenty five percent state of charge will also reduce life span.
According to Thoughtco.com, as the temperature drops, the current available also is reduced. Especially in lead-acid batteries, this turns the Peukert effect into a fairly detrimental situation. Your device requires more current, the lower temperature reduces the current, causing the device to "demand" more, which likely drops the voltage, increasing the current demand. It's a downward spiral.
You could consider alternative options. Dedicate a battery to a low wattage heater that will keep the primary battery warm. Especially insulate the enclosures for the battery or batteries. With the low temperatures involved, a fairly large expanded polystyrene cooler with thick walls (more than one!) nested together could give you the thermal stability required to complete the tasks.
Connecting batteries in parallel will provide more current to the device, but will still suffer from the cold temperatures.
Lithium chemistry based batteries are not to be charged at temperatures below freezing. If your camera contains such a battery and the supply voltage is at a charging level, you may be damaging the internal battery as well. These batteries also suffer from reduced output and capacity in extremely low temperatures. Our EVs present a dash panel warning at low temperatures indicating that regenerative braking may be reduced. The onboard computers prevent the system from charging the pack during extreme cold, reducing the chance of damage to the pack.