Rather than trying to design a better cup, what if we bypassed the cup altogether? What if the incentive to use a reusable cup was so strong that fiber cups fell out of use?
At participating restaurants and coffee shops, customers who brought in a reusable beverage container would pay less for their drink than if they used a fiber cup. This would hold true even if the reusable container was the size of a kiddie pool.
Viability is the obvious concern here, but the promotion wouldn't run indefinitely. It would run for a few years in order to get consumers into the habit of bringing reusable containers into restaurants. Hopefully, even once the promotion stops, the behavior would continue.