I would think, that the Babysitter, has her own rates, which she then tells the Client.
Overnight Sitters, usually get paid more versus daytime Sitters.
At least in my State that is. AND overnight Sitters, the ones that I know of... do NOT pick up or drop the child off themselves. The parents... do that. They do not "transport" the child, unless they have a car seat themselves and a contract, that authorizes transport of the child etc.
It doesn't matter if the child is sleeping, the Sitter, is STILL ON the clock and caring for the child and their safety etc. No matter what the scenario is. And they don't often get to sleep, themselves.