Dye won't weaken the material, but bleach will. Paint won't work at all. It will take a lot of trips to the dye bath to turn gold to blue - especially if it's bright blue.
Your best bet is to get new webbing. Many send their belts to Snake Oil for restoration, but their webbing is very thin. I don't recommend them. Check with your local auto upholsterers. One is bound to restore seat belts or at least know of someone local who will.
