Is there an exact constant value for the speed of light in a vacuum distorted by a gravitational wave? In a vacuum, construct a cylinder of photons arranged as follows:

A 'measuring apparatus' computes the speed of light as 299,792,458 m/s.
However, suppose the vacuum is distorted by a gravitational wave

The 'measuring apparatus' now computes the speed of light as 299,792,457 m/s
because it takes light longer to travel along its geodesics.
Question: Can any 'measuring apparatus' truly compute the constant speed of light
since the underlying geodesics can never be fully known?
Notes: The example above uses Gravitational waves, but the Milky Way Galaxy core rotating (anything affecting spacetime) is equally viable. Re-calculating $c$;)
 A: The problem you pose in this question is, I think, quite a general and interesting one: we have a definition for a physical quantity and we wish to measure it, but the definition assumes "isolation" from the outside world, whose influence in practice will never be precisely zero.
This is an experimental problem, and it is "solved" only by iteratively understanding the sources of error in the measurement with ever-increasing accuracy.
As far as this goes, gravitational waves are a very small concern when measuring photon trajectories: our best interferometers
are able to detect GWs, sure, but besides being custom-built for the job they operate in a heavily noise-dominated regime; seismic, thermal, quantum noise have a much larger amplitude and it is only through clever data analysis that we can measure the GW signal.
So, any lightspeed-measuring (or rather, meter/second measuring) device need not worry about GWs as a noise source.
That said, you still might wonder whether the previous discussion is warranted: couldn't a certain GW configuration be distorting our measurements arbitrarily without us being able to tell?
It might be possible, sure, but within the theory of relativity the equivalence principle comes to our aid: if we look at a small patch of spacetime it will bear an ever-closer resemblance to flat spacetime, therefore the theoretician can be happy to tell the experimentalist to just build her measuring device to be small enough.
Then, they can be sure that gravitational effects can be neglected.
A: A quick trip to Wikipedia and a look at Speed of Light will tell you that the speed of light is now, by international agreement, defined to be an exact value. Formerly, it was not exactly defined, so that had implications for the definitions of other units of measurement, particularly the meter.
Definitions of measurement units have changed over the years in keeping with the technology available for minimizing uncertainties in derived measurements. The International System of Units or SI system was recently updated in 2019 based on current  state-of-the-art in measurement technology.
For the average person on the street, and even in most industries, the update affects practically nothing, but allows high-precision industries to reduce uncertainties and improve tolerances for advanced scientific and engineering research and development.
A: 
Question, can any 'measuring apparatus' truly compute the constant speed of light
since the underlying geodesics can never be fully known?

To start with although the velocity of light in vacuum and the velocity of photons in vacuum as far as our measurements and theories go have the same value, light is composed out of photons in a complicated quantum mechanical way, but photons are not light, similar to a "bricks make up a wall, but a brick is not a wall".
In the theory of general relativity all zero mass particles follow geodesics with velocity c, a photon is a zero mass particle.  If in trying to measure the velocity of light in vacuum a gravitational wave disturbs the instruments and/or the light beam, a different velocity of light may be measured, as happens with light in a medium. The individual photons making  up the beam still go with velocity c, follwingthe geodesics, according to the theory.
The theory of General Relativity has not been falsified up to now,  it is even used to put the satellites in orbits and even the GPS system depends on GR.
