First and foremost Welcome to FT and I apologize that nobody has answered your question.
My experience is with Lido. I have not been to the Moulin Rouge.
I went with my parents, not part of a group tour and were placed in the worse table possible. My father took out a 200 franc bill and gave it to the head waiter and told him this was a special evening and he wanted a better table. And voila, a great table appeared.
200 francs was probably around 30 dollars and there were 3 of us.
I would slip somebody 10-20 euros if you dont like the place you are assigned.
I know this does not answer your questions but.....whatever (I am sure there will be some comments about this).