There's no way you wouldn't be diverted to an airport that doesn't require a Cat III landing, even a IIIA is stupid dangerous for a trained crew. iirc you only have a 1-2 second correction window for any III landing where the ILS cuts out, compared to something like 8-10 for a I (which is plenty of time for an amateur that's been coached on the procedure to do a missed landing then give it back to the autopilot, that's super easy). Plus the number of airports that can even do a Cat II at least used to be pretty low (150 in the US?), and III-rated airports are even fewer (I dont think there's double digits that can do a IIIb, and afaik there's still none in the world rated for IIIc...)