I think the corner of the inner border should be like the other squares and the outer border plain.
I agree that the binding will "finish" off the little white square in the outer border but I don't think it needs the little white square. It tends to draw my eye away from the beautiful blocks and fancy pieced inner border. The final border is the frame for a beautiful quilt and gives the eyes a chance to rest before being drawn back into the beautiful blocks and pieced inner border.