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. Like 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 II (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)