> Anecdotally I've heard of examples where (e.g.) washing china plates has a greater carbon cost than using disposable ones.

Only if you use a dishwasher. IF you do dishes by hand, the only cost is water and soap. Strictly speaking from an environmental point of view the time you spend doing it would be considered a "gain" instead of a "cost", given that you will be probably not engaging in modern electronic entertainment for the duration of the task.

In an industrial environment, such as a restaurant, the cost analysis would be non trivial, given the more complex supply chains that businesses have.