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*[(n1n2)*dL/dT+(dn1/dTdn2/dT)*L]*delta_T
 w1 is the frequency of the field @ 1064 nm (radians / sec)
 L is the length of the crystal
 n1 is the refractive index of PPKTP @ 1064 nm
 n2 is the refractive index of PPKTP @ 532 nm
 delta_T is the temperature fluctuation at the crystal
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.
 If this is true, any suppression of temperature noise should be seen as a reduction in the phase noise of the green when the Mach Zehnder is locked.
