That's horrible! And there apparently is no easy way to fix it. How sad, how infuriating, how disheartening! And no matter how careful you are, those things happen, almost as surely as an open-face peanut butter sandwich lands peanut butter side down on the floor. My sympathies to you !!!!!!!!!!!!!!
As to avoiding the mess, I remember an elderly friend who after many, many quilts said she figured out a way to avoid it most times, and and she decided that most of the time was better than nothing . . . . Anyhow, she kept an old yard stick on hand that was actually 4' long (does that make it a 1 1/3 yard stick? is there a name for a 4' yard stick???). When she was finished laying out the sandwich and pinning it from the center out to the edges, she very carefully put the yardstick under the edge, inched it toward the center, and then moved it side to side, and "felt" for any resistance. Then she kept going all around the quilt checking for any area that "felt" different from her end of the yard stick. Only then would she begin basting, and she used giant stitches -- about 8" -- from the center out to the edges, and then a smaller stitch -- about 4" -- around the edges. When that was finished she very loosely rolled the quilt, unrolled it showing the backing side up, and if all was good she re-rolled, unrolled with the front side up, did the yardstick probing one more time, and only then did her regular basting. The pins came out only when the regular basting was in progress.
I have other friends who go to their churches, shove tables together in the community or all-purpose rooms, get out the masking tape, and literally tape the backing to the table tops as the first step in making their sandwiches. They say it makes a huge difference for them in avoiding shifting fabric, and gets the sandwich up to a comfortable height to pin and then baste. Since the entire process takes the better part of the day, they call the church office to choose a time when nobody wants to use or move the tables.
I have decided that for my next large quilt, I will make the sandwich upside down, so that the backing is what shows. My theory is that a pieced top has a better chance of staying put, as will the batting since it "sticks" to all the piecing seams, but that the lighter weight backing is what is most likely to shift. And I always hope no friend ever stops by to laugh while I'm doing a sandwich on the floor, but when I get the sandwich laid out, before I add any pins I open a new case of soda pop in cans and use them as weights, which I believe (but cannot prove) helps hold everything in place while I do the initial pinning.