I had another thought on batteries, but didn't get around to posting yesterday. I had started buying the inexpensive batteries proliferating at Amazon and eBay; I think this started with a recommendation/link from an online photo guru (or jester, depending on who you ask), but can't swear to that now. Even initially I wasn't convinced the cheap LR44's were lasting well, but over time I've satisfied myself that they are a poor bargain. I also had a bunch 2032's, or similar, that are used in car fobs and musical instrument tuners. These were branded with a well respected electronics manufacturers name, but performed very poorly compared to Eveready, etc.
I've been using Energizer 357 batteries and getting much more satisfactory performance. These are a silver battery instead of alkaline; that seems much better in general though I don't know specifically about cold weather performance. They are cheap as 10 packs from Amazon.
There are a lot of factors here, including age of the battery, but I'm not interested in being a test bed for products, so am sticking with the batteries that have been most reliable for me.