Isn't there a hotel in the airport?? Or am I thinking of Miami International? (We stayed there on our way to a cruise.)
We did the exact same thing you said. We flew in late to MCO and yes, DME did take us at 12am. Anyhoo, we went to the resort we were staying in and just paid for the night and then our package started the next morning on a new reservation. It was perfect...we didn't even have to change rooms, although sometimes you may have to. We ate breakfast and were in the parks early...it also saved us money on airfare to go the night earlier and stay over than traveling the next day so it was a no brainer.
I don't advise staying in the airport, though...too uncomfortable and NOT safe at all. Save yourself the worrying and get a cheap room somewhere. Get a room on International Drive or at a value on property...much safer and you'd have a bed.
Good luck.