If reference noise is a problem, one could replace the 1N4579 with one of those low noise Chinese 2DW232. They are even about the same voltage, just the ref. current might need adjustment via R864 to get the current for low TC. There is a thread one this ref. in the metrology section.
A first test could be just to measure the noise of the used reference. Maybe this type of zener is also very low in noise - at least it uses a higher current than the LM399, so the potential is there. Though a good reference, I have not found anything about it's noise specs.
The long time reference should still be the LM399 reference, so it would not make much sense to use an expensive LTZ1000 here. Using a separate short time and long time reference would make sense if the short time one is lower noise - than it can be a good solution, as one would need an internal calibration / adjustment from time to time anyway to compensate for drift in the integrating resistors and amplification to bring it to 10,x V.
The reference contributes to the overall noise in two ways. First is the obvious, that the measured value is proportional to external voltage divides by ref voltage. So relative noise level of the reference will also appear at the output. Extra time used for AZ and zero phase might amplify the noise a little. This can be a significant part at higher numbers. As this is low frequency noise, it is hard to impossible to improve by filtering.
With many of the ADCs (also the K2001) there will be a second effect, as for zero reading they use +ref und -ref to compensate in a kind of PWM style. So noise from that modulation frequency band (e.g. kHz range) can also contribute, especially for near zero readings. Here filtering from C821 might just starts to be effective. So a larger capacitance here could also help a little to do some more filtering. The LM399 is specified at about 100 nV/Sqrt(Hz) in the kHz range - so it could be a significant contribution (e.g. more than most other amplifiers). However with a significant voltage to read, the LF noise should be the more important part, especially with slower readings.