"The problem here is the different parts of the aircraft become supersonic at different times..."
When I read this I thought, what parts, and where. There was an interceptor the Convair Delta Dagger where this was a major problem. Google for "Area Ruling Supersonic Flight" for some info on this. I think one Geoffrey De Havilland met a nasty occurrence of this when it made the DH 108 he was flying, break up in mid-flight. It certainly made flying the Hawker Typhoon a dicey experience at altitude, according to the histories.
If Lockheed know what they are doing, it need cause no more damage than localized heating: if they don't know what they are doing - and at times that is the impression they give - it's "goodnight nurse" for the pilot.