We don't use a plastic bowl for the washing up,far too much fiddle.
Our (large) kitchen sink is stainless teel, and seems to hold the heat quite well.
When we were in France last year we bought a plastic(cushioned) mat with lots of small rubber sticker type feet.This attaches to the floor of the sink and make a cushioned base for the china,or anything breakable to sit on.It's useful too, if like me you are a butterfingers and dropping things into the sink!