is there any requirement to measure whether the durations of 9 192 631 770 periods of different primary frequency standards and/or of the same primary frequency standard in different trials, had been and remained equal to each other, by (presumably) unambiguous means (such as the "ideal clocks" described in MTW §16.4) ?
There's no explicit mentioning of such a requirement that I know.
However, if some unambiguous means of comparing durations (such as the "ideal clocks" described in MTW §16.4) were taken into consideration then the comparison of durations of oscillation periods allows(1) to conclude
whether any one given instance of an oscillator (such as a "primary frequency standard") had been constantly "perturbed" (even possibly including having been constantly "un-perturbed"), or variably perturbed; for whatever expected or unexpected "reasons", and
whether or not any two given primary frequency standards had been equally perturbed (even possibly including having been constantly "un-perturbed"), or not; for whatever expected or unexpected "reasons".
Related considerations are conveniently presented in response to issues raised by this answer which had been posted earlier, from which the following quotes were taken:
how [...] to implement this
There's no strict requirement to actually implement an "ideal clock" according to MTW's description;
but merely to judge and quantify (or even only to estimate) how the relation between given setup participants differs from having constituted such an "ideal clock", to be "corrected" as suitable.
(Similarly there's no strict requirement to actually implement "caesium atoms unperturbed by black body radiation"; but the requirement presents a definitive ideal relative to which the given primary frequency standards should be "corrected".)
Still it may be asked which sort of observational data might be the basis of such a quantification at all(2). Looking at the illustration of MTW Box 16.4 I'd think foremost of the (possible) appearance (or disappearance) of "interference patterns" involving the relevant setup constituents.
external assumptions [...]
Suppose that you could implement an MTW clock: you launch into orbit two mirrors and a caesium clock. [...] The light and the atomic oscillators should therefore be completely in step.
(To simplify the discussion: let's say this is found during an "initial setup phase" of at least $n'$ caesium oscillations.)
Suppose, further, that for some reason after a number of orbits (probably quite large) the light in the cavity is no longer in step with the caesium atoms, and does $n$ round trips for every $n' \ne n$ caesium oscillations.
First to note regarding "orbits" is that the MTW prescription (as quoted in the excerpt) involves certain necessary "precautions", or (arguably) "corrections". If $n$ represents the corresponding "precise" number, and correspondingly $n' \ne n$ caesium oscillations were found during the "trial phase" then:
the mean oscillation frequency of the caesium atomic oscillators had changed in comparison to the "setup phase"; it had been "perturbed" differently in the "trial phase", in comparison to the "setup phase".
How can you distinguish that conclusion (i) from the alternative interpretations that (ii) the distance between the mirrors changed,
Certainly the (relevant, "cautious" or suitably "corrected") tick duration of the orbiting MTW clock remained constant; as a matter of definition.
Of course, the tick duration(s) (or "ping duration(s)", or "signal roundtrip duration(s)"), either as "corrected", or "uncorrected" ("raw"), may be taken as measures of "spatial separation" between relevant setup constituents;
typically (for formal distinction from all sorts of other durations) with some fixed symbolic non-zero prefix attached, such as "$c_0$", or "$c_0/2$".
or (iii) the speed of light did?
It is certainly absurd that a mere and supposedly fixed (non-zero) symbol such as "$c_0$" should have changed from "setup phase" to "trial phase"; at all, and especially by a real number value "$n / n' \ne 1$".
you can accompany your setup by a ruler (i.e. an actual ruler made of atoms)
But relevant is surely not just any actual ruler made of atoms, but only such actual rulers made of atoms for which the "spatial separation between its two ends" (or "between two relevant marks") remained equal to a significantly better ratio than the real number value "$n / n' \ne 1$".
So how to determine which of all given or even all imaginable rulers satisfy this requirement? (The SI "metre" definition should give a valuable clue.)
The definitions (idealized thought-experimental descriptions) of how to measure "duration" and "spatial separation" are of course human conventions.
But there are very explicit and useful guidances on which of all imaginable conventions are preferrable, namely Einstein's assertion:
All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more material points.;
in fulfillment of Bohr's requirement:
{W}e must employ common language {...} to communicate what we have done and what we have found.
in practice [...] us humans, and everything around us, are made of atoms, and therefore [...]
... therefore we may want to determine their possible "perturbations", trial by trial, even if we had not expected them, and even if we have not yet pinned down their possible "reasons".
Notes (added after the initial posting):
1: Utilizing the notion of "ideal clocks" as described in MTW §16.4, a concrete definition of a duration unit (fittingly called here one "artefact second") might be the following:
"The artefact second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atoms of the NIST primary frequency standard, referred to being at rest, at a temperature of 0 K, starting at UTC date January 1st, 2000, 00:00:00 .
This definition refers specificly to the artefact second being distributed in reference to the Marzke-Wheeler procedure."
2: That is, only as far as it is considered impractical to gather observational data which is explicitly required for carrying out the Marzke-Wheeler procedure, and thus to identify a suitably dense network of "ideal clocks"; including explicit coincidence determinations such as at the tick event preceding event $\mathcal{ C }$ in the first illustration of MTW box 16.4.