if you want border to match as you describe you you have to adjust the width of the cut and folded binding accordingly.
The old "rule" was to cut binding 6 times the width of your seam allowance. However, this didn't allow for folds and loft of your binding. If you fold your binding and measure in thirds, then 1/3 should be the width of your seam allowance. Try this and adjust accordingly. Cutting at 2.5" and using a 3/8" seam allowance is a bit too large. 2"-2 1/4 is more fitting.