What a beauty! I've been sitting here for the longest time just admiring that sleeve. I'm easily distracted by sparklies.
I don't think re-backing would accomplish anything. The irregular shape would make the process a nightmare and wouldn't lend much more support.
I would be tempted to cover the backing board with black or dark green velvet. (Would that cause a conservation issue? I don't know.)
If the customer is thinking of this piece as art, I'd mount it flat; if it's being preserved as an article of clothing, it might be nice to put a layer of quilt batt under the velvet to add some dimension.
Of course you're planning on lots of tiny stitches accross the width of the piece.
The sleeve looks heavy and if the lining fabric is fragile it won't be able to support it's own weight. Installing the backing into the frame on a slant might help.
I'm looking forward to seeing the finished product.
Kit