Do you have a data sheet for the power supply (or supplies, if you want to put both the brick and PicoPSU inside)? You need to calculate the efficiency and then you can work out the heat output over time.
You would then do a "power in" in watts minus "power out" in watts and assume all of that "useless" power is dissipated as heat.
If I take the original Duo supply as an example, the input power is () 1 Amp at 100v (assuming that the biggest current draw is at the lowest accepted voltage) and the output is (full load) 1.5Amps at 24V.
That's 100W in and 36W out, dissipating 64W into the case, which is 64% efficiency. (this is probably quite far from what it actually generates)
Assuming you're generating multiple voltages you would need to do these calculations separately and add them up.
From there you need to know how much heat the Duo brick can dissipate, which I would model as a sealed box, something like:
https://www.powerstream.com/temperature-rise-in-an-electronics-enclosure.htm
Putting in some assumed values for the Duo brick (I measured one but I've not got one to bits to measure the wall thickness):
Assuming it's made of ABS, the walls are 2mm thick and the size is 115x65x30mm in a 20C room the supply would raise to about 50 degrees C, with a case temperature of 30 C. ABS is generally considered safe at this temperature, and whilst it would feel warm it wouldn't set anything on fire.
That said, do please check my maths before burning your house down.