Get the Doula! Doula's have been proven reduce chance of (expensive) C-sections. The doctors and nurses, even if they are the best in the world, work for the hospital, not for you. They will make decisions based what is good for the hospital. Check your hospitals C-section rate. The WHO expects a C-section rate of 10% but most American hospitals are closer to 30%. A hospital will push people into C sections to reduce their liability.
The doula works for you. They will help you make decisions and support you to avoid an unnecessary C-section. If you end up in the percent that needs one, it is what it is, but trying to reduce that risk is totally worth it. Also a doula will often include birth and breastfeeding education, so you may be able to skip those classes and save money there. I'd recommend getting a doula sooner than later, because they will often offer pregnancy support also, and they charge a flat fee, so the sooner you get them the more for you get for your money. My doula also cleaned our house, made meals, did 6 loads of laundry and took photos for us. They're really a super deal.
You say expensive C-section, but depending on your insurance, there isn't an added cost to the consumer. I met my OOP max if I had a vaginal birth based on the prenatal care and hospital stay, nothing cost extra for a C-section for me.
(My C-section was due to a partial abruption though, so a doula wouldn't have done anything to prevent that. Our hospital midwife clinic has a 8% c-section rate, and that's who I go with, and the midwife called in a surgeon when she determined I would need one.) A doula, on the other hand is expensive and out of pocket.
Doula's around here charge a flat fee for labor/delivery services, but most charge hourly for the other services you describe. Between $30-$50/hour for prenatal and postpartum services, night services cost extra.
It's about $1,000-$1200 for labor/delivery support, which generally includes 2 half-hour prenatal visits and one or two hour-long post-partum visit.
I did nearly get a doula for my second birth (due to the PTSD from the stillbirth), but after interviewing, decided it would be way too much money, and the support offered didn't seem like that much more than what my skilled birth partner could provide.
I agree, book the doula early. Many are booked 6 months before a due date.
So much depends on how your hospital does thing. The midwife on call never has more than 3 patients (and there are OBs all over the freaking place- when my alarms would sound, like 10 would rush in); the L&D nurse is dedicated to you; he never left my side. If you hospital is drastically different, I can see why having a doula would be worth the money