Turns out that no - at least not for any extension cord of practical length and design.
Copper is a great heat conductor, but not nearly as effective as a dedicated heat pipe. You'd have to run it along the length of the cord to get a temperature reading above noise.
Our supplier advertised a sensor, which we found eventually on the other end of the PCB. The reaction time would have been five or ten minutes if at all, essentially the whole device would burn down and eventually it would take the sensor with it.
Turns out, the sensor would had to sit directly opposite the FET on the PCB, and there had to be a plugged via going straight through for sufficient thermal coupling. A lot of the physics boils down to simple aspect ratio, and normal vias are really thin.
(The real solution would have been putting the sensor on an FPC, and to glue onto the transistor. The real real solution was to make sure the part didn't blow up in the first place, which was much easier to do.)