I would stick to your current plan. Rent the car in Koln. It will have winter tires and may have chains. All cars in Italy must have chains.
One way rental from Germany to Italy can be very expensive.
Trains are fine if you want to be on the tourist path.
You need a car to get off the beaten path, which from Koln to Italy is so very beautiful and interesting. Stop and stay in small to medium sized towns and you won't be disappointed. People are warm and prices reasonable.
For one way rentals,check out Kemwel.com

In December it mostly rain, but at higher elevations there can be ice and snow. Oh, I see you are from SoCal.
But it's for you mother, do whatever she says. I can guarantee you that in the future you won't regret it. And she is right about Europe in general and Austria in particular, you do need a car.
I go every year, mid-Dec to mid Jan.

Good luck