Enter "A", who recalled his little-boy Lego storage—a "parachute" bag, he called it. He described it in detail, and it sounded like the perfect solution. I got to work, with my adviser close at hand for some grommet work.
The idea is simple—a circle of nylon (though any fabric would do), grommets around the edge, and a cord. It lies completely flat, so the girls can work from the pile of pieces in the center, eliminating the likelihood of having them scattered about the house. When it's clean-up time, they simply pull the strings and they're done!
The girls are thrilled to have their new bag, "A" is happy to have re-created this bit of his childhood, and the soles of my feet are thanking me.