If I am understanding the problem, than the many different bias pieces are causing the quilt to stretch as you sew the borders on? I don't see any way to avoid it except applying some kind of fusible to the back of the whole quilt (pricey). I think I would go ahead and quilt it without borders. I would then re-square the quilt and add my borders Quilt as You Go style. The quilting will stabilize the quilt and then you can add the borders.
QAYG- You lay the border right side down on the quilt (pin). You lay the backing piece right side down to the back(pin). You put a strip of batting pinned to either the front or back piece and sew the pinned sections on the top edge of the quilt.You press all of them outwards and then do the same to the bottom of the quilt. You then repeat the process on the sides and then finish quilting the borders and bind. Good Luck!