I wonder if a vacuum drive belt would do it. Some of them are small enough that I think they would stretch over the wheel. It might be possible to find rubber bands wide enough, or perhaps cut some cross-sections from a bicycle innertube to stretch around the wheels. If you use an innertube, wash the talc off.
There are also urethane drive belts / bands. Not sure how hard it might be to find any of an appropriate size.