I keep a 2AA light in my car. I keep it loaded with a pair of NiMH rechargeable duracells, and I top them off every once in a while. I also keep a four pack of AA lithiums in the glove box as back-up batteries.
Both NiMH, and lithium primaries do fairly well in the cold, at least quite a bit better than alkalines do.
A few years ago, I bought a pair of rayovac AA NiMH batteries with a USB charger. The batteries are in a remote, and the charger is in the car. It will do AAs or AAAs. I have a couple of AAA penlights in the car too, and they have been charged a few times with that charger. The problem with a 3 cell AAA light is that the batteries don't have a lot of capacity, and most chargers want you to charge them in pairs.
At any rate, I'd probably fill it with NiMH batteries, and keep some lithiums as spares, as I do for my AA light.
Another possibility is to use
18500 LiIon batteries, they might fit. I measured a 3*AAA carrier and it is 54 mm long. A 18500 battery is 50 mm long
Panasonic NCR18500 3.7V 2000mAh Rechargeable Li-Ion Batteries
LiIon batteries also handle the cold well. The batteries that I linked to are not protected, and I wouldn't use them in a light that has a parasitic drain like your light does, because you could run into an over discharged LiIon, which could be dangerous. You might be able to find some quality protected 18500s though.