Playing with a ballistics calculator can give you that answer. For this example, let's ignore whether or not temperature affects powder and primer burn. Here's a couple of examples punched into JBM that I just snagged from historical weather near here. I punched in Eley Tenex at 1085 fps with 10 mph crosswind. These are results at 100 yards with a 50-yard zero.
Then below the two real-world local weather examples I've also punched in some unrealistic examples where I only changed the temperature. While the other parameters are also going to be different as the temp changes, looking at what happens when you only change the temperature can give you a basic idea of the type of thing that'll happen as the temperature drops. As you say, when the temperature drops the atmospheric density goes up, and the bullet drag along with it, and this results in more drift and more drop. Never looked at a list of examples like that before, but I have had to adjust my silhouette zeroes for turkeys and rams by up to ~1/2 MOA or more in the cold/hot extremes. Never had those NWT cold extremes here on Vancouver Island, though. Yikes! hehe.
As to whether or not the temperature affects powder or primer burn, I'm sure it does, but I don't know to what extent. I do know that in the summer months a lot of us would get by ok for silhouette with some cheap CCI Standard Velocity ammo. But once it starts cooling off that CCI lube starts getting pretty hard. I'm more inclined to blame the lube than the powder, but maybe it wasn't burning very well, either. The performance of that stuff when it starts getting cold just tanks and starts looking pretty ugly out there at 77 m and 100 m for turkeys and rams as the SD and ES goes crazy. I actually started bringing extra hand warmers just to keep the ammo warm. One day I even tested with a few of them jammed in between the scope tube and the top of the barrel over the chamber area to keep the chamber warm. With the barrel/chamber kept warm and the ammo kept warm the stuff shot as well as it did in the summer months with the SD and ES smartening up again, as much as the CCI could. haha. I could swear I had some CCI chrono data from that testing, but I can't find it now. I did lose ~50 strings of LabRadar data at one point, so maybe that stuff was among the ones I lost. The CCI isn't great ammo, but taking the summer months data as baseline it did perform a lot worse in the winter months, and it did look pretty much the same as the summer month data when the hand warmers were in place keeping everything warm. I do have a few strings/targets with both Eley Club and Eley Club Biathlon with my silhouette gun from August of this year, I think, that I can compare to once it starts getting below freezing here again. See whether or not the Biathlon stuff behaves more as the temp drops. I'll have to remember to do that once it's a bit snowy here.
I don't imagine the various ammo manufacturers are making biathlon ammo just for pure marketing reasons. There's surely some performance differences once you drop below freezing that they're trying to make up for. They're loaded a little bit hotter than the other stuff. And some of them use different powders and lube, too, which they say are more suited to those temps. I'd say the hotter loads are to help deal with the denser air. I don't know if that's necessary, really. But powders and lubes that behave better in the colder temps make sense to me.
