I don't know of a nylon lined stable sheet to recommend, but just wanted to say I've invested in a 'full size' Bossy's Bibs RugGuard, which you could maybe consider. It's not cheap but I think it's a great product.
In my case I love it firstly because it prevents rug rubs and hair loss. And secondly it's brilliant as it keeps the inside of rugs clean. As and when it becomes necessary, you could very easily wash the RugGuard, dry it and take back to the yard the next day - much cheaper and easier than to keep stable rugs clean. (Obviously doing it like this the rug's outside won't get washed, but to me that's not nearly as important as having the inside of the rug clean as that's the area which is in contact with the horse's body.)
I had one from Shires, a tough stable cotton sheet on the outside and an independent nylon lining. It was FAB. Not seen them lately, not to say they don't still sell them. They were red.