Here is a "math teacher" suggestion for using your requirement of the charm square aspect: sew two charm squares together on the diagonal a la HST-style; press to set the seam and trim leaving a 1/4" seam allowance. Next, draw a line where you want the parallel seam. Now, using the desired "trim triangle", position it along that line, right sides together, so that when stitched it will complete the square.
Note: the "trim triangle" will be a bit larger than needed so you could be "approximate" in placing it, if you are willing to square up the piece after adding that triangle. There should be a minimum of trim waste and the second "trim triangle" can be saved to be used in another project. Actually, all of the "extra" triangles could be sewn together to make HSTs.