First of all…great name.
I had the exact same problem with a contact sensor on my garage door (Western NYS, just across the lake from Toronto). It wasn’t that the button battery died, the device didn’t get enough juice from the battery to even run. I would be surprised if the thing lasted a day before i conked out completely.
Also, any PIR motion sensor is going to have a tough time in that type of cold. You’re beater off going with a wired and mains powered contact sensor on the doors than a motion sensor. IMO.
I have a multisensor in my garage freezer to monitor the temperature. I have had it in there for nearly a year on the original battery. It dropped to 60% pretty quickly and I thought it would need to be replaced right away but it has held in the 60% area and is still working.