So I pulled the rope back out, which rather boggled and distressed him, then grabbed a broomstick and tied the rope through the hole in it. Then I arranged the rope for easy unwinding without tangling, put the end of the broomstick into the hem, got all lined up, and then pitched the broomstick down the hem, javelin style. It made it most of the way to the other end in one go, and then I could scrunch and pull the broomstick almost a meter at a time, getting it out the other end in just a couple of operations.
The look on the director's face was priceless. On the one hand, he was thrilled the wretched job was done. On the other hand, I had utterly shown him up.