I talked to the passport agency, and was told that once it is two weeks from my scheduled departure date (which I have to prove with an airline receipt), I can schedule an appointment to apply in person at the passport agency in Chicago and they will hand-process my application. They said this would take one or two days, and they would send the passport to me through overnight delivery. So the whole process should take three or four days at most.
This whole thing has been (and will be) a tremendous hassle, but at least it looks like I will be able to get the passport in time. Even if I do have to take off work to make the 8-hour round trip drive to Chicago in order to get it taken care of.