Going back to a 1D heat transfer model, and matching Stephen's numbers for the area and length of the thermal strap, I confirm that the conductive power of a single strap is indeed heavily constraining the cryocooler capacity. For my simulation the peak power for a 0.5m copper strap with area 6.71e-5 m^2 is 1.06 Watts with an average of a few hundred mW during the cooldown.

The main difference with respect to Stephen's numbers is that I account for temperature dependent conductivity and heat capacity, include a radiative sink (surrounding vacuum tube at room temp), and take the strap to be made of RRR 500 Cu.

Attached is the predicted temperature at the end of the strap as a function of time when the operating point of the cryocooler is set to 123 K. Note the cooldown delay caused by the single half-meter strap (with respect to the cryocooler cold head).