Are you handy with tools? I would scavange the construction sites for a big piece of wood to use. It would most likely be chip board, but you could glue the cork right onto that.
You can primer the other side and put the chalkboard paint over that. For a more rustic look, several 2 x 4’s would make a cute plank chalkboard. You could paint each piece before assembling to that all the sides would be covered with the paint and look more uniform than make a frame for around the outside and paint it the a color to compliment the room.
My 4th grade teacher made slates for all of us to use in class with plywood and chalkboard paint.