I delivered at Gilbert Mercy, but our pediatrician was not associated with that hospital. So instead of finding a new doctor, we just used the house pediatrician that was on staff and just made an appointment with our current pediatrician within a week after our daughter was born.
I'm assuming you already have a pediatrician with your son, so I'm not sure why you are being asked to find a pediatrician with privileges at Scottsdale Healthcare North. They should have one on staff. You should be able to use your own after your daughter is born.
I used to have United Insurance and was never required to have my pediatrician use the same hospital that I delivered in.
And just a note that is a great hospital if any complications arise. An old co-workers wife is the head nurse for the NICU. You will be in good hands.