You really need to find out if it is the battery or indeed the starter motor, or even a bad connection such as the engine earth strap - which can show up intermittently.
The symptoms you describe do point to a weak battery, and you always have to suspect battery first when problems start with cold weather. If the car would start with jump leads to another vehicle, then this would certainly point to battery. If it still struggled, then that would point to the starter, or a bad connection somewhere. Any starter motor can fail, especially if it has done plenty of miles of short journeys, but it's wise to eliminate the cheaper options first.
I think the CVT auto starter is different to those fitted to the manual, but any decent auto factor should be able to supply a recon unit from the registration mark. Alternatively, there are plenty of used ones about - there are a few different part numbers used, but I believe all the mark 1 auto starters fit the same.