Hi, I hope they did not send you a /6 seat, as that was what happened when I ordered from BMW Bayer, I had to send it back to get it changed for a /7 seat.
The /6 seat is a couple of inches longer and hangs out over the tail light which prevents it from seating properly. (pic attached)
The bayer part no. on the underside of the /6 seat was 5253061
The /7 seat is shorter and should just barely clear the tail light.
I don't think there should be any difference between the 1980 and 81 rear sub-frame, so would be surprised if this is the problem.
If you are having trouble getting the hinges under the frame, there is a knack, slide the front hinge under the frame first and then maneuver the rear hinge under the frame with the seat in the down position, once both hinges are under the frame open it up to locate the hinge onto the pins.