If you re-glue the stitching protector strip to the tire, it shouldn't be a problem.
When you repair a tubular (as people used to do all the time), you pull a section of that strip off to get at the stitching, open the stitching, pull a section of tube out, patch the tube, put it back into the tire and re-stitch the seam. Then you re-glue the strip back to the tire. So it's been done many, many times. I used to use the same adhesive to glue the strip to the tire I used to glue the tire to the rim and never had a problem.