Setup:

I calculate the conduction and radiation from my SolidWorks assembly to figure out how much power our cryostat needs and how long it will take to cool everything. 

All units are metric.

There are 4 seperate elements with (a constant temperature within the element) and 7 conduction pathways. The 4 elements are the breadboard+beamsplitter mount (labeled "bread"), the inner shield (labeled "inner"), the outer shield (labeled "outer"), and the vacuum chamber (labeled "bath"). The 7 pathways are the following

bread+bath: conduction and radiation

bread+inner: radiation

inner+outer: radiation

inner+bath: conduction

outer+bath: conduction and radiation

I use SolidWorks and a little bit of analytic calculation to figure out the conductances. 

After calculating the coupling coefficients, I set up 3 ODEs to get the time dependence. I have some code to give me the exact time it takes to reach 123 K. From the equilibrium results, I calculate each pathway's power and the total power lost to the bath.

Results:

At equilibrium, 4 W are lost to the bath. This is the minimum power required to cool the beamsplitter, but we are going to need more to cool it in a reasonable timeframe.

If we go with a 25 W cryohead, which I think is very possible between 123 and 294 K, it takes nearly 8 hours to cool the beamsplitter and breadboard. It takes a little over a day for the inner shield to reach equilibrium and 2-3 days for the outer shield to reach equilibirum.