I think I have found the answer to the doubling phase noise after a few pages of algebra and differential equations. I will have to confirm this at a later date, but for now I assume it's correct:
When I lock the phase of the Mach Zehnder in the IR, the phase of the green is:
Phi = Constant + w1/c*[(n1-n2)*dL/dT+(dn1/dT-dn2/dT)*L]*delta_T
In summary for the "locked Mach Zehnder scheme"
If I treat the phase of the IR as my error signal, and push it to zero (or to some constant), the fluctuations in my green (and thus sensing noise), will be dominated by temperature fluctuations of the oven.