Why can't you use your machine? The only part that needs to fit under the needle is the binding, which is pretty skinny. I think many, if not most machine stitch it to the front, turn then hand stitch to the back. I machine stitch both sides. You can machine stitch to the back, turn it to the front and stitch close to the edge. You can stitch it to the back, turn it to the front, and use a decorative stitch. You can stitch it to the front, turn it to the back, then pin or glue baste being sure to cover the seam, then stitch in the ditch from the front. My favorite way is to use a faux flange:
Flanged Binding