So I've done it - added several more lines of quilting, and what looked like a rumpled, creased failure last night is now a quilt that I like both look and feel of. It's now washed and pegged out on the line - the weather today is providing me with a natural tumble dryer, as it's very warm and breezy.
This has been a useful learning experience: it was so much easier to do the extra quilting, with the quilt completely stable and with no pins in it. As I was using a standard domestic sewing machine, it also made a difference to have lost the 4in or so of backing and batting that had been trimmed off each side. And my batting shed slightly, so I was glad to have it out of the way and not leaving fluff on the quilt. I think I'll try doing this again, deliberately next time - do enough basic quilting to stabilise the quilt, then add the binding before finishing the quilting, especially if I want to add any more elaborate flourishes.
Thanks for all the help!