Okay, after leaving it running for 3 hours with a thermocouple in it and the "grid" (variac) set to 280Vrms I've found the following:

-->The air temperature just above the transformers reaches about 65 degrees Celsius
-->Laser gunning the metal-can electrolytics reveals they are at about 75-77 degrees Celsius (which we can approximate as the PCB temperature).
-->The temperature was still rising even towards the end of the test.

This is what I can do to 70 degree Celsius PLA material with one hand by the way:

So, I know that the PCB is it's not convecting effectively enough (or likely at all) in the new case. However, I also know that the heat build up is so slow that ANY air movement will cool the thing down.

Well, then aside from ice cubes, air-conditioners and desk fans, this has to happen:

It's basically an add-on case assembly that clicks on top and adds active cooling to the device. It uses a 60mm 5V USB fan that will simply plug into the computer that is measuring it's output. In this way, the power is being drawn through the existing computer an is not utilising a potentially noisy additional power supply that would have been connected to the same power outlet as the Grid-2-Audio measurement device. However, I am still very concerned about induced EMI noise from the fan itself. We'll see how that goes soon: