FIGHT FIGHT FIGHT
I've got fantastic coverage under CIGNA, but you have to figure our how to work the system.
Do not expect the hospital billing department to continue to submit this billing. They will usually only do it once or twice unless you are able to get an authorization number from CIGNA. Keep in mind that it could be a billing error in the way it is being coded...if it happened at 03:30 in the morning, I assume it wasn't scheduled! That makes it an emergency and then the rules change regarding your coverage. If you were already planning a c-section, it might have been accidentally coded that way. Remember that the billing department just enters what gets sent to them.
Now that I've got that out of my system...how far you get with this may depend on which CIGNA plan you have. HMOs can be really difficult...but with PPO or OAP you probably have out-of-network benefits that will pay at least a portion of the bill (after deductible).
Last...don't be afraid to call back multiple times. You would be amazed at the different results depending on the person you get on the phone.