Your normal rug cleaning shampoo should be sufficient for cleaning fringes, unless there is a problem stain that will not come out. Bleach, peroxides, etc. and cotton fringe are an area where it can get away from you, and you buy your customer new fringe for their rug.
MOST fringes are not ultra white, cotton is not that color unless it has been bleached.
You need to educate your customers not to expect the fringe to be as white as their tightey whiteys....