I really like the doll quilt idea.
What I would do is keep all the seams associated with the back in place when cutting away the front and sleeves. This would keep the knitted back intact. I would probably hand baste the quilt backing in place to make sure everything lies flat. Then I would *carefully* machine stitch using a long basting stitch all around. If everything looks good, I would go over that with a shorter machine stitch to add strength. Next step would be to mark the cutting line using a ruler and Sharpie permanent marker. I would stitch stitch the binding on before cutting, to ensure that the cut knitting doesn't come apart or interfere with the sewing. Next-to-last step would be to cut along the marked lines. Last step would be to turn the binding and hand sew to the other side.