AC coupling doesn't solve it, that doesn't disconnect the scope probe ground clip from the chassis or from the safety ground.
All AC coupling does is block the DC component of the signal, which is not the problem here. The spurious current here flows in the ground path, not the probe-tip signal path.
I see their scope has two probe inputs, "channels". If they used both, put one probe on each end of the resistor, and left the ground clips hanging, they could subtract (or, often, invert one then add) channel 2 from channel 1 and ideally see the real voltage across the resistor without providing this massive path to ground. Tying the two ground clips together might reduce noise somewhat. To get clean signals to subtract nicely, you might have to tie the grounds to one semi-isolated almost-floating point, with finite impedance to ground but large enough to block any spurious heating currents. You can temporarily connect the two probes to the same point to ensure that the gains really match (adjust until they do) and the subtraction really results in zero. The loop formed by the two scope probe cables will pick up EM noise, so keeping that small helps--perhaps loosely twist the probe cables around each other.
The setup as apparently used could easily add 20 watts to the test device chamber through the ground connection, without any visible evidence. But if the device keeps heating while the scope ground clip is disconnected, then this cheat was not being used.