There is absolutely no reason why a woman cannot labor naturally with a big baby. In fact, millions of woman do it all the time, all over the planet, without needing c-sections or pain medicine. In America, we are c-section and epidural crazy. The thing with epidurals, however, is that they numb you so much you can't push, and then you end up under the knife.
I labored naturally, with no pain killers, etc. for over 24 hours. It's all mental. If it is something you really want, it's yours for the having. All I know is that I left the hospital the next morning, and while tired, I felt great. I even took my baby for a walk the next day.
I took a Bradley class, and I'm thankful for doing that. Since you are due so soon, I'd recommend renting some birthing movies. I found it helpful to watch tons of women having babies, naturally, mostly at home. The representations we get from Hollywood are just wrong. I'd also highly recommend getting a doula, especially if you aren't using a midwife.
We were born to labor. There is nothing to be scared about. And while painful, you don't remember any of it, and trust me, I would do it again a million times -- and am doing it again in December.
Just make sure you tell everyone that you want to labor naturally and that you do not want to be asked if you want an epidural. If those around you are supportive, you will have no problem achieving your goal.