Knitqueen, could you possibly cut out a piece of sturdy cardboard to place in the bottom? I'm not sure if that would work or not....or maybe even a piece of firm plastic. Whatever you used, you could cover with a thin fabric (something very "drapey"), by spraying the piece with spray adhesive before covering it with the fabric.
Or, you could even knit a piece (square or rectangle) from the same yarn, plenty big enough so that once you've felted it, you could cut it in a shape to match whatever bottom piece you chose (cardboard or plastic), then glue it to the top of that piece.
I'm not really sure how much sense I just made, but it made good sense in my head.