Never mind. I managed to pry up the plastic on the other side and jam a box wrench in there. I wedged it between the nut and the bracket then applied as much sideways force as I could to the wrench while turning the bolt up top. Eventually I got it.

It's a 4-sided nut on the other side and it is supposed to be welded to the bracket. Now I'm wondering how to put it back. I wish I had a tiny arc welder although I could probably never get the nut lined up correctly anyway.