When stopping on sites with washing up facilities we usually make use of them as the sinks are bigger and it saves a bit of water.
I used to use the washing up bowl to transport the dirty dishes and SWMBO kept saying we could use something better.
Eventually I succumbed and bought one ot these :-
Ortlieb 10L Bowl
It does the job very well and also has other uses such as washing your feet etc.
This site ( All Outdoor ) was the cheapest I could find.