I think the you should be fine on Pragel Pass on a bicycle. As far as I remember the restriction is more for motorised vehicles.

I've done the climb from the Klonthalsee side. The climb is actually quite nice especially when you get to the pretty much car free bit. The descent down towards Schwyz is pretty steep, narrow and twisty compared to alot of other passes. Also much of it has thick tree cover which can make the surface quite slippery in places. So my advice would be to take it easy!