You definitely cant use cushion tires outdoors. Lowering the pressure on pneumatic tires will cause damage and instability. Also, low tires tend to scrub and twist as they roll, making things worse. There are soft compound tires with no lugs, that are slip resistant in wet conditions. We use them on Tennant scrubbers. Soft wide footprint, but dont last long.
To protect floors, use plywood. We drove a 10,000 lb Reach truck through a Museum in Ottawa that had beautiful marble floors!
thick plywood was laid down as the truck moved through sensitive areas.