In my view, studded tires would be more important than fat tires, if I was forced to choose between one or the other.

As for batteries, the main difference would not be performance, but range. Regardless of the temperature, the motor controller supplies the needed current to the motor, and you're limited to 20 mph (or whatever is the upper limit for power assist) anyway. I haven't ridden an e-bike, but I know a bit about electrical control systems. Regarding temperature, bringing the battery inside would make sense, and while riding, the battery heats itself.

An issue with cold weather is that plastic parts tend to become brittle. This depends on tradeoffs made during design.