I added a copper bar that goes from the coldhead to the breadboard which holds the beamsplitter. I also added a much more realistic model of the cryocooler with a variable lift (the amount of power than can be removed from the bar as a function of temperature) After modeling different coldheads and layers of shields, it is quite apparent the time to cool to 123 K doesn't really depend on how much shielding there is unless the cryocooler lift is barely more than the power lost to the bath. This makes sense analytically.

Depending on the amount of shielding, there are between 4.3 W and 8.7 W lost to the bath at equilibirum.

Attached is a table with different coldheads and levels of shielding. {}:{}:{} means that coldhead couldn't cool the beamsplitter to 123 K. Note: values might not exactly line up due to changing parameters. I have some uncertainty to picking an emissivity depending on context. Luckily this level of accuracy isn't needed.

Next steps are getting quotes on the thermal shields and coldheads. We should get a coldhead with at least 20 W of lift near 123 K.