I was originally thinking it would be good to stitch through the middle of the tag, but then realized that if you went all the way through, you would have a random diagonal line on the front of your quilt. If you can manage to do a running stitch just into the batting below Scarlett's name, that would be ideal. Otherwise, my opinion is that it's just a label -- if someday it starts to get baggy, then you can always unsew & replace it.
Another option to consider is using a small amount of fusible web or Fabric Fusion (permanent glue) to fuse the fabric in a few places to the backing. If you do that, I'd make sure to run it in a very straight line -- or apply all over -- because otherwise you'll end up with pulls and puckers.