Having had experience with tie-dyeing in the 1960's , I can tell you getting colorfast dyes out of materials is next to impossible, You can lighten the color but getting it snow white probably wont happen.
Using Rit color remover will sometimes lighten a color enough to be dyed a darker color. Going from light to dark is pretty easy....going from dark to white , is tough. Be careful with bleach , like Clorox , it will damage the fibers if too strong a mix is used.
Try to find one or get the white webbing and install your hardware on it to create a white sling.
White slings should be available as they were used in parade dress / drill teams etc. , might try Numrich Arms Gun Parts Corp. for them.
Gary