Cold batteries won't start rockets
I could be entirely wrong, but I think the cold at 80,000 ft will provide more of an ignition problem than the low pressure and that, while freezing the rocket motor may be an issue, keeping the ignition battery warm enough to deliver enough amps may be the biggie.
Sacrifice an ignitor in order to measure the current it needs if the supplier can't tell you this, and then set up a second cold soak test to make sure you can keep the ignitor battery warm enough for long enough to deliver the required current after LOHAN reaches maximum altitude.
Size the battery to have enough capacity to keep itself warm inside its insulation for, say, twice the estimated time needed to get to 80,000 ft and still retain more than enough kick to goose the ignitor real good. I think you'll find that keeping itself warm needs more capacity than firing the ignitor and that you'll need a thermostat in the heating circuit to minimise power drain at lower altitudes. The PARIS time-to-altitude numbers should be a useful input to this capacity calculation.
Li-poly batteries don't work at all well below 25 degrees C and most other types get discouraged below zero.
Maybe a supercap, kept topped up by a lower current battery with chemistry that can deal better with lower temperatures, would work as well as a Li-poly kept at 25C. A look at the specs hints that you could charge it on the ground and forget about the top-up battery, though a top-up battery might be useful to counter any current leakage if you get condensation on the supercap and wiring as the rig gets cold. Supercaps may be lighter than Li-poly batteries (I can't find any weights), are certainly cheaper and are unlikely to be bigger.