Assuming a vaginal birth, there are some lower-cost options than paying out of pocket.
1) a birth at home or at a birth center with a midwife typically costs about $3000-5000, all expenses included
2) some hospitals will charge you less if you have no insurance and/or
3) you can sometimes prepay hospitals and get a reduced rate, because they are guaranteed to get the money with no hassle, no billing, no filing insurance, etc. (just make sure you have all costs figured in; an epidural alone may cost $2000-3000 for the anesthesiologist and all fees, though perhaps you can get a reduced rate if you want to prepay that as well)
It sounds like your estimations for paying the insurance and paying the hospital are about the same, so you could start saving the $800 a month that maternity coverage would cost and use that to pay for the birth, as well as shop around at different doctors, midwives, hospitals, and/or birth centers, to see what the actual costs could be for a cash-paying customer, especially if you pay in advance. Even assuming that the best rate you can get is $10,000, if you could save $800 a month for a year (9 months of pregnancy plus 3 months of "trying to get pregnant"), that would be almost the total amount you would need to pay for the birth.
Another angle is to look at your current expenses and get serious about cutting out extra "fluff" from your budget (you'd probably have to do that anyway, to set aside $800 a month, but maybe you can get even more serious than that), and see if you cut your expenses to the bone, how much you can save. At worst, you'll pay off some of your debt, and at best, you'll realize that you can pay a lot more on your debt than you thought you could, and/or that you can save a lot more than you thought you could, and you'll be able to pay cash for a birth.