You may try this: put a three-speed freewheel on the rear hub. If you want to use another gear you loosen the axle nuts, move the chain on another cog of the three available. Then you re-align the wheel, retension the chain and re-tighten the nuts.
The chain line might not be too straight but in a way you have re-invented Campagnolo's very early Cambio Corsa, without both the quick release and the shifting arms. It will work with a freewheel pack only, of course.
And you may also need some kind of device mounted over the chainring on the seat-tube that keeps the chain from falling off. Those fork-like thingies existed in the old days when there were no FD to do that job.
BTW, there are no fixed multi-speed 'cassettes'.
Let us know if you've tried it out.
Edit: The 17-22-37 rear might not be workable. The wheel would have to move back and forth way too much to accommodate for the chain-slack. I'd rather have a 17-19-21 or similar.