Considering that all they really need to do is run another copy of the exam through off the printer and that you are hardly looking at a full room in a bridging exam I can't see why they should not try to acommodate you, as the mistake was theirs... shame you have to take the trip out there in time for the exam on Monday, though, on the off-chance they changed the schedule or something and can't sort it out over the phone.
I would spend the weekend practising being polite but firm...
I hope this is enough encouragement...