Winter commuters sometimes switch between studded and nonstudded tires using the wheel switching method you describe. It is less trouble than swapping just tires.

In practice you will probably find you switch alot less than you figured at the start. Lights and fenders needed for a commuter don't provide much value on the trail either.

You can get a new trek 820 cheap to commute on for not much more than what you will spend on wheel, tires, and cassettes. Used beaters are even less and make good commuters.

Dan