I'm not an insurance expert. However, one thing to realize is that the OB is paid one lump of money for everything they do for you and the baby (the global fee). The hospital "privileges" the OB, allowing them to use the facility, but neither charges nor collects from the OB. So, from the hospital's perspective, they can only bill you for services rendered (perhaps in 2017, perhaps in 2016).
I doubt you can convince the hospital to "pre-bill" for the delivery - it seems like it would be impossible. The OB may be able to bill after the delivery, but they often collect up front, and refund later. It is hard to collect from someone who has already taken their baby home, and they don't want to invest months of prenatal care in someone without assurance they will be paid. I would expect that the OBs office manager would be the appropriate person to authorize later billing, though they will likely still require a deposit. The global fee should include the charge for delivery, if its vaginal and perhaps if it is an uncomplicated c/s (for the OB - remember, none of this is going to the hospital.)
Typically, one does not pay a copay with prenatal visits. Also, there is another thread about "double billing" - you will likely have a deductible for your child due in 2017 as well (for the post-natal care).