I don't think it would really take AZ-OPs in the control loop around the LTFLU. They tend to cause more trouble due to EMI issues going out from the OPs. RF signals originating from the OPs could effect the reference reading and the level of interference could be effected in a hard to predict way by thinks like attached cables.
At least for the OP for the voltage loop an normal OP (like OP07, LT1097,ADA4077) should be OK, as only something like 1/100 the OPs offset would appear at the output.
For the temperature reading this is less clear. Still 1 mK corresponds to about 2 µV at the sensor. So some drift may not be that critical.
However the temperature compensation part could be rather sensitive to the OP. R22/R20 gives a gain of some 100, while R7/(R24+R25) give an attenuation of some 700. So about 1/7 the OPs drift would appear at the output, which is not a good ratio, though probably just acceptable.
Another possible problem could be resistor drift from R20,R21,R22,R24+R25, especially if R20/R21 are not adjusted to keep the current through R22 and R24 low. The high values resistors tend to be more drifty than others. Chances are R20 and R21 also would need to be low drift, not as important as R7, but still with quite some effect.
I would try adjusting the zener current, if less transistor current is no longer an option. The Zener current should also effect the TC, though less predictable.