which uses a CR1220 coin battery like this, with ≈37mAh:
Which they say will work for about 5 years.
Will it work if I replace that battery (by soldering wires) with something like this?
Or 2x1.5v AAA batteries linked together in series.
And if yes, what is the maximum lifespan of the battery I can get? Because they degrade over time anyway, regardless of the capacity and I will keep the battery also in cold weather.
I think two AAAs would work well, but I don't know how they are affected by cold. And of course you would need a two-AAA battery holder. I think the CR2 should work very well, with no worries about cold, but you would have to find a holder for it.
Another option would be a CR2032 (235 mAh). You can get little fob holders for them, and wire that in:
If you haven't already bought the Adafruit module, you might consider a similar version at much lower cost. I just bought one of the round modules here, but have yet to try it out.