It's been a while since I was in college, however if the OP is booked on a 6AM-ish flight the "flat-tire" allowance is only going to cover him until at best just before 9AM. I never knew an exam to be scheduled before 8AM. That gives the OP less than an hour to take the exam, get to the airport, get in line and present himself to an agent to plead his case.
My suggestion is to try to get the youngest agent you see, or one who looks like a Mom who could have college aged children herself, at the counter who hopefully will be sympathetic to your situation. Be honest and tell the truth. Airport agents have heard every exaggeration and blatent lie in the book and they can usually tell if one is not being honest. I believe honesty will get you further than deceit.