Boy Racer,
When considering how to clean the fabric, remember that the fabric was used to allow a wet harness to drain and dry. Depending on the condition of the fabric (over time the heat and cold, wet and dry cycles take a toll and the treads grow brittle so would never consider physical adjuration). I would use a detergent like Dawn mixed in hot water and sprayed it on the harness. I would then rinse it with and water spray and allow to dry. If you want to take a chance, I would use a small amount of washing bleach in the washing mix to brighten some of the lighter colors but I would first test on a small section. Keep in mind that the threads of the woven cover are either strong enough to sustain the wet/dry cycle or not and, if not, the cover (not the harness) will need to be replaced. I would never expect it will look new.
Good luck,
Ray(64BJ8P1)