Maybe I'm missing the obvious because I have very little experience with machine quilting, which I'm trying to learn right now, but your mention of using the walking foot makes me think you may not have tried free motion on it with the feed dogs lowered or covered. I have the intention to create a little project with calico on the pieced side and fleece on the back, and I will soon find out what's what, I have a feeling! If that doesn't work well, maybe it would be possible to quilt it with some paper or other tear-away product under it? Alternatively, could it be worked with the fleece side facing up? (It would be impossible to follow the patchwork, of course, though you could probably mark the corners of squares with pins or wash-away marker.) If it was not sidetracking from my present project, I'd run over to the machine to test a bit right now. Good luck!