It is doable, and its crazy to to say this but the best strategy is similar to car shopping. Go to several hospitals and get a quote. Then you go back to the each hospital and ask them to beat the best quote.

It's an usual approach to childbirth but it's been done. Hospitals, for almost any service they provide, do not have a stanardized cost. Prices can vary dramatically. The last person I heard who did this paid almost $10k. Initial quotes from the hospital they used before "shopping around" was almost $20k. The variable would be complications during delivery. medical issues/needs and/or premature birth. Regardless, best of luck!

Again, it's uncommon but doable.