The question on the software version is a very good one. The is a possibility to reduce the observed type of noise by a kind of digital filtering, especially for the 7 and 8 digit modes. As this type of noise could be a principle weakness of the design, newer Version might use this.
IC201 is not expected to contribute much to the noise, and only the more white noise type. So it does not make very much sense to upgrade here. If you really want to, the more direct, upgrade should be a LTC1052.
The more promising candidate for an upgrade would be IC202, as this chip is expected to give a contribution of the right type of noise. For a test one could also just reduce the bandwidth of the existing LM301 by modifying the compensation (e.g. increase value of C204 by a factor of about 5). This is more of a test only as this could have a negative influence on linearity.
More promising would be the extra amplification between the integrator and the comparators - many modern DMMs use this trick to improve on the comparators. A limited bandwidth here would also reduce noise from IC202. This modification is expected to reduce a noise source of the observed type (mainly higher frequency).
A totally different point, might be looking at the supply. "noise" from the supply could influence the comparators, as the decoupling in this area is not very good, but hard to fix / improve. Even if not visible on the scope ripple on supplies could cause trouble and might be the difference between good and noisy meters.
One source for supply noise are the switched mode regulators. So looking at there output caps or tests with added caps there might be worth a try.