I have no idea if this will work for seat belt fabric but I soaked my '84 Sea Ray boat's interior with interior fabric cleaner and then use my pressure washer to blast the dirt out of the fabric webbing. I first tried soaking and then scrubbing and that was somewhat tedious and not really getting the job done. The pressure washing or actually rinsing worked best. I was hesitant at first but what the hey it's a BOAT interior that has the possibility of getting wet sometime in its life.

Before and After with interior/fabric cleaner soak and pressure washer rinse: