I've ridden fixed through two seasons of ice and snow.

If you only have clearance for 20 some cm tires, go with a Continental 4 Season Grand Prix in front and whatever you want in back.
The back tire doesn't matter.
If it gets so slippery you can't go, you can't go.
I ride an Armadillo in back because I don't like flats.

Gear down so you remain in control without using your front brake for anything anytime.
It the rear tire slips around, it gets exciting.
If the front tire slips around, it gets painful.
I used my brake once and the front tire went out from underneath me.
Four broken ribs, a punctured lung and a month without work.
Gear down.

I ride 53X18 and 53X17 in the summer, for 77 and 82 gear inches respectively.
In the winter I go down to 53X19 for 72 gear inches, and I can easily ride without a front brake, although I always have a front brake for the unexpected, like a chain coming off.

If you can fit 37cm tires, go with Nokian 106 studded tires.
They will last forever and forever and forever.