OK, now for plan C (or are we up to D or E?). What if first, before adding the triangles around the edge you trimmed the backing & batting flush with the folded edge of your background fabric. Then instead of triangles, use squares a bit larger than the ones in the center of the quilt. Before inserting them, turn 2 adjacent sides in 1/4" and press well. Now put the raw edge corner into the window, stitch as usual, then turn the loose part of the square to the back & hand stitch down. This will bind the edge and give you an interesting look on the back too.
Hope to see a pic of the idea that works best!