Stabilizer might make it easier to do the satin stitch but I'd be careful of what is used as it could become "stiff" and not as cuddly for the little person. This is such a cute work--congratulations on it and the new baby! Bumper sticker sounds like an excellent idea. Maybe a heart there?? Or could just take it as a bit of patina or a dent in the bumper? At least, it's in the right place for that.