...
For a 10m pathlength of acoustic interrogation we'd see a travel time difference of driven by change in average sound speed. Take that change to be half the gradient, so 0.15K.
That induces change in sound speed of 0.6m/s/K * 0.15 ~ 0.1 m/s. So difference in sound travel time is 10m/330 m/s vs. 10m/330.1 m/s or around 20 microseconds.
We can easily resolve this acoustically, it's 10X our measurement error.
The light traverses the primary-to-secondary z distance not once but 3 times, converging on the third pass. So thermal path is really more like 20m so requirement is perhaps 2x more stringent?
We should set up two parallel acoustic monitors, parallel to the optical axis on opposite sides of the mirror. This will measure differences in the line-integrated index.