As with everything there will be a contract in place between the DfT and the MoD which should resolve many of the issues you mention. Wattisham and RM Chivenor BTW operate under the Military Aviation Regulations regardless of what company provides the ATC and RM Chivenor is now part of the naval estate. Yes, it has no ATC but it does have an ILS, which belongs to the MoD not the DfT
Exeter will be too close to Lee on Solent and too far from the IOS and the South West Approaches. Culdrose has all the support facilities needed, plenty of Hangarage (none available at Exeter without significant cost).
You could put a SAR site anywhere you liked to be honest, even St mary's. The issues are flexibility in poor weather, facilities available, including runways, engineering, met, ops, flight planning, communications, navaids, ATC or not and location, location, location.
Culdrose, I am afraid, ticks all of those boxes (intended!).
(Don't live next to Newquay do you and hope for a SAR job without having to commute to Hellstown?)