When I had the problem of the truck not getting up our steep driveway after an April snowfall, my Dad suggested lacing short lengths of rope through the wheels.
I wouldn't use stainless screws because there's no benefit for the cost; but why introduce a metal object that will fall out eventually and possibly cause a tire problem for somebody else "down the road"?
There
is an automobile ice-racing class that allows bolts that go right through the tire tread.
If you have tubes in those snowblower tires, that may be an inexpensive option for you—
but there's no going back!