I would do it in blocks, maybe 10x10 or 12x12. Take a piece of fuseable interfcing, block size, and lay it glue-side up. Do your weaving over that. You have good "pin technique". Then press according to interfacing inst. Let it cool. Now, put your top together and your quilt sandwich. Using a zig-zag or decorative stitches quilt over your strips. Securing and quilting-all in 1 step!
You can do it "whole cloth", just get a lot of interfacing (the lightest one you can find) Can't wait to see it!