I have been almost exactly where you are. I used pre-made binding as the armhole facing for an unlined dress for myself once and it was so scratchy and uncomfortable that I couldn’t wear it.
Could you cut some cotton lawn or silk on the bias (or some of the fabric you used for the blouse) and hand stitch it to the seam allowances of the piping? I guess that only deals with the allowances though and not the piped edge. Would covering the allowances help to put some distance between your skin and the piped edge though so that it wouldn’t rub?
Or could you unpick it and re-do it with some home made piping? Sometimes some well placed stay stitching and some starch can hold things in place for you to re-sew, even if you’ve clipped curves etc. already.