Hi,
Well it is unclear what you are using for the heater (nichrome wire in a coil perhaps) and ambient temperature.
Also, will the tests be continuous or intermittent, thus giving more time for everything to come back to room temperature.
Heating problems can often be estimated by using electronic circuits with resistors and capacitors. I would hope you have some exposure in this area. The heater would be a voltage or current source for example, and the inner box a capacitor, and the link from inner box to outer box another resistor where the outer box is another capacitor, and the link between outer box and ambient another resistor and the ambient would be a sink. So what we end up with is a low pass filter where each stage node is a temperature. The inner box capacitor voltage represents the temperature of the inner box, the outer box cap voltage represents the outer box temperature. The capacitor values are based on the specific heat capacity of each box, the resistor values are based on the thermal resistance of each box. So the inner box heats up, some heat conducts through the walls to the outer box, then some heat conducts through the walls of that box to the ambient, and the ambient is considered constant temperature.
Does this view make sense to you? Granted it may take some measurements to get the parameters estimated.