I think the SR-71 would be closer to the Concorde than the shuttle as far as dealing with the thermal effects of supersonic flight. The Blackbird grows several inches during high speed flight (but then the thing is 100+ ft long to start), leaks fuel (JP-7) on the ground, and has corrugated surfaces on parts of the wings, all due to airframe heating.
With the Concorde and SR-71 the issue is one of prolonged heating (hours) while the shuttle deals with extreme heating but for a bounded period (a few minutes).