US Customs require the paperwork for 3 days prior to your exporting the vehicle out of the States. This is presumably so they can check the VIN numbers to see if the vehicle was stolen.
So, you arte stuck with staying over or making a return trip.
There are storage lots, I believe, in Blaine that you could check into and leave it there for the 3 days, but I have no names or numbers for these.
The other thing to check at US Customs is the hours of operation of the export approving guy. Used to be only one person working 8:00-4:00, Monday to Friday. Miss those hours and you're sol. But call the US Customs Office at the border crossing you intend to use to check and confirm hours. This is no hassle thing, as long as the car isn't on the hot sheet. When I brought mine across he didn't even go out to look at the truck.