I don’t understand how you can plan to do a manual landing from a Cat III approach. We can only plan to do an automatic landing from a Cat III approach. See
Getting to grips with Cat II / III Ops
“2.2.4 CAT III OBJECTIVE The main objective of CAT Ill operations is to provide a level of safety equivalent to other operations but in the most adverse weather conditions and associated visibility. In contrast to other operations, CAT III weather minima do not provide sufficient visual references to allow a manual landing to be made. The minima only permit the pilot to decide if the aircraft will land in the touchdown zone (basically CAT III A) and to ensure safety during rollout (basically CAT III B). Therefore an automatic landing system is mandatory to perform Category III operations. Its reliability must be sufficient to control the aircraft to touchdown in CAT III A operations and through rollout to a safe taxi speed in CAT III B (and CAT III C when authorized).”
Hence the 2 autopilots must have separate power supplies.