hmm....you know, now that I think about it, I never went back to school the fall semester before Nicolas was born....and they covered me until the birth and Nicolas until 6 weeks after. Maybe you could call the company and find out if they will cover you continuously, especially since your EDD is so close to your birthday. I mean, we're talking about an extra couple of weeks max, you know?
As for your midwife, she probably won't want to bill for the birth before it happens, just in case she has to do something extra. Are you planning a homebirth? In the unlikely event of a transfer to the hospital, you would be stuck holding the bag for the hospital charges. If you're planning an unmedicated hospital birth....well, there's all sorts of things that could happen then that are impossible to plan before hand. I'm not trying to scare you, really I'm not, it's just that birth can be unpredictable.
I think your best bet would be trying to talk to the insurance company first. If they won't budge, find out if your midwife takes your state insurance. What state do you live in?