It's fine to piece the last border, but if you don't like it that way then your plan will work too. If you do piece the last border, it would be best to put a line of stay stitching around the edge so that it doesn't stretch while you're sandwiching it. Also be aware that after you sew the first inner border your quilt dimensions will change, so the number of HST's and plain squares going around it will be affected, and it will also be harder to match those blocks to the blocks in the quilt, if that's a consideration.