Depending on the size, you could cut a piece of foam board the same shape. It's very light weight. OR in the piece above, I would add a second sleeve at the top of the wide section and insert a thin dowel. might want an additional dowel at the bottom of the wide part, then I would add some weight to the bottom. Could use drapery weights or a larger dowel, with yes another sleeve.
Great quilt, by the way!!