We do overnight shifts occasionally.
We have a set 10 hour shift 9PM-7AM it the same hourly rate.
Verbally clarify that you sleep and wake up with the baby to either bring baby to be nursed or you will bottle feed and the mother sleeps, and then you and the baby go back to sleep.
I had a client that misunderstood and thought the doula would be up all night during the overnight shift: cooking, cleaning and doing laundry. And was upset and called me to know why the doula was asking where she could settle in to rest during the night.