Personally no amount of money would make me work later. I am already open from 7:30-5. I think you will regret it if you accept this family.
I had a mom begging me to open early for their child. Offered to pay extra! Nope! Those are my hours and that is that.
I really think a child coming in right at nap time is going to have trouble adjusting and wanting to nap. So not only are you working extended hours but your nap time will be a struggle