I've been running the 1399 in my 34401P with 820ohm (10ppm/C) resistor.
The 1399s are usually by aprox 70-100mV lower than the 399s.
Now - with "RC" you talking the "snubber" I assume, I also experimented with "RC low-pass" in front of the AD706 there (finally I decided not to have the RC there).
Without the snubber I saw oscillation like "telegraphing" with 1399 as well as with MAC199 in my "noise meter" in past, so I would place the snubber there for sure. I have not seen an effect of the snubber with the 399.
Another thing I observed in past is following - any wiring too close or direct to the 1399's solder joints at the bottom pcb side increases the noise fluctuations. Therefore I wired the 1u MLCC (it should be 25V one - mind the decrease of its capacitance with voltage there) via piece of manganin wire with aprox 5-6ohm resistance.
Btw., in the last 3-4 months I opened/closed my 34401A box (incl. removing the inner shield with the glued temp sensor) aprox 120 times..

So be warned..

PS: the "typical 30-40 seconds" in our discussions about sliding/rolling standard deviation mean the "rolling window size in seconds". So you have to measure the stuff much longer than 60secs. I would rather go with 20-30secs window for 34470, btw. The sliding std mean you buffer the last 30secs of the incoming samples and you calculate the std over the entire window upon each new incoming sample.
Or make the ADEV.
From my experience the replacement of a 399 with a good 1399 - the 2uVpp one in my case (my 4 samples had 1.7-2.5uVpp noise in 0.1-10Hz thus the spread is large) - the decrease in the noise is almost immediately visible at the incoming numbers - the std @10PLC went down from 1.9-2.0uV to say 1.4-1.5uV, the replacement of the OP27 with OPA140 in the integrator pushed it down to 0.8-1.2uV.
At 100PLC you have to see something like 500nV std as the mean in the sliding std (with OPA140).
Always do the measurements with 10V input (10V range).
That all measurements were against my ADR1001#1_10V which itself is rather noisy.