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Start by adding an illumination system to your bike, and see if you're comfortable with it. I built a 12V system for mine. It's a 7AH sealed lead-acid battery from a UPS, a toggle switch, a red trailer LED lamp, and a 3W LED headlight. It's kind of overkill, actually.
Bicycle snow tires exist, but I will not ride if there's snow on the ground or in the forecast.
The cheapest thing to do would be to just use the Corolla. Three short, cold-start trips a week aren't going to cost you that much. Use 0W20 and a block heater if you want to minimize the impact of cold starts.
If you want to build an NEV, just say so. It'll be another vehicle to insure and maintain, but it would be pretty cool, and EV conversions have been done on budgets smaller than yours.
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