I finished this project this evening. I'm rather pleased with it, and although I didn't plan in the beginning to stray quite this far from the original, I rather like the outcome.
I used Pellon Peltex 72x double-sided fusible for the shaping, and covered both sides with the chosen lining fabric. I then trimmed one side of the lining to the size of the fusible and turned the other side over it and stitched. This created a thing rather like a very sturdy placemat, to which I attached pockets and then zigzagged the side edges off the edge, so it's kind of like a flat-locked seam.
I had planned to use straight bamboo rod handles, but they didn't work with the very stiff shape, and so I tried a lot of different shapes and ideas, and came up with the D-ring bamboo shape upside down. I attached the handles with 2 lengths of turned tube each, stitching the ends before attaching the knitting to the lining.
I sewed the 2 knitted pieces together on 3 sides, then handstitched the top edge of the lining to the appropriate point on the knitted pieces, and finally handstitched the knitted facing to the inside.