Tl/Dr: We make corrections as we always do. They are relativistic corrections, but they're just corrections. Barycentric Coordinate Time (TCB) and Geocentric Coordinate Time (TCG) demonstrate precisely how we get around the nightmare of relativity
We have fought this battle before, on Earth. So we can see some of the important parts you need already built out. Unfortunately, some of it will require sending atomic clocks around, or at least information sent by radio waves.
This is the journey of atomic time. Time is simple, until you have to deal with timezones... or you get down into the peculiar worlds where physics gets tough. This is where the International Astronomical Union (IAU) lives. And what they deal with is truly mind bending.
And I am sorry in advance for how long this answer is. The raw amount of detail behind our current equivalents for GCT are astonishing. There's good reasons we do the things we do... we just do a lot of things. The first half of this answer is the very generalized "what is a time system anyway, and what are they for?" That paves the way for decisions we've made with time systems in use today, and can be used to project towards a fictional GCT based on how the system is being used. And honestly, this information is spread across so many documents that having it in one place is of some use. I've broken it up into 3 sections: background, history, and solutions to your problem
Background
What makes a good time system? In fact, what makes a time system at all? The Time Ontology in OWL captures a great deal of what we care about. It builds off of TemporalEntities, the Instant and the Interval. These are the instantaneous events and extended events that we might be interested in speaking to. From the perspective of a single observer, these Instants and Intervals have a well understood ordering.
Why start so abstract? Well, in the end, time systems exist for a purpose: to assign mathematical values to things, and that can be pretty darn generic. The Aztecs did not need our precise mathematical definition of time, but they might care whether their ritual events (an Interval that might last a day or days) was OverlappedBy a solar eclipse. As it turns out, the latest studies show that their calendar wasn't precise enough for predictions, but they did care about the order of events enough to write things down. It is these events that can be measured which matter.
To quantify these, we add two additional concepts: the time point and the duration. These are used to quantify Instants and Intervals and their relationships. Time points are associated to where Instants occur, and durations are associated with the extended span of Intervals. A duration is the span of time between two time points.
And this is where it starts to get a little quirky. As it turns out, while its easy to define a duration as the time span between two time points, practically speaking durations are easier to measure. If I'm judging a pinewood derby, I rarely know that a race started at 2024-02-18T13:45:13.125Z and ended at 2024-02-18T12:45:21.439Z (putting the cart before the horse and using ISO 8601 date-time stamp notation). That kind of precision is hard for timestamps. However, with nothing more than a stopwatch I can measure the duration associated with the race Interval is 8.314 seconds.
Why is this? Durations are repeatable. If one has an experiment which reliably takes the same duration every time, we can use it as a clock to determine how long other things take. However, a time point only happens once. We can't do any repeated measurements to quantify its time point (we can do parallel measurements, but nothing after the fact).
As a result, nearly every time system ever defined in human history first defines a duration. Whether it's the duration of the day or the duration it takes a standardized incense stick to burn, durations come first. We then add an epoch to define time points. An epoch is an Instant with a pre-defined time point associated with it. Once one has an epoch, one can calculate the time point associated with any instant as long as one can identify a series of adjacent intervals and their durations. All practical time systems invented by humans take the form of a defined duration (day, second, incense-stick, etc.) and an arbitrarily chosen epoch time point. The duration can be re-measured and re-defined, but the epoch cannot be re-measured.
That last bit is important, because it opens the door for a key property of time systems. We can often define the relationship between two time systems. With such a definition, we can compute a time point in one system using another. If we know the relationship between Mountain Standard Time and Eastern Time, we can compute that 6:00 PM MST is equivalent to 8:00 ET (they are associated with the same set of Instants). You will see we leverage this property heavily in our time systems, and it will be very important for dealing with relativity for GCT.
So that's the introduction. What a long document this is turning out to be! Now let's get into the actual time systems we've used in the past.
History
We can skip over an astonishing array of local time calendar systems. The old systems are quite interesting in that they are practical. Time was measured to support the time operations that were needed. For example, in many cases time was measured differently at night because the primary thing needing timing support was shift changes.
This all changed with the invention of the chronometer. For the first time, a clock could be used to accurately determine longitude for ships at sea. But the clocks were not as immaculately precise as our current atomic ones. If left free-cycling, they'd soon drift uselessly out of date. So they were tied to observations of something rigid and unyielding: the rotation of the Earth. The first universal time system, Greenwich Mean Time(GMT), was built to this need. GMT defined an epoch of "noon at the Royal Observatory in Greenwich" and all clocks were set from it.
And I already lied. It actually wasn't set to noon. Noon drifts over the course of the year due to effects like the elliptical orbit of the Earth around the sun. GMT is Greenwich mean time because its associated with the average(mean) noon over the year, and was actually measured from studying the angular distance between the moon and certain stars at night. So we see a common pattern that will emerge over and over: using mathematics to compute a time system whose events are not exactly experienced anywhere.
GMT caused some confusion. Astronomers started their day at noon, which means a nightly set of observations all occurred in the same "day." Most people started their day at midnight, so their daily activities all occurred in the same "day." This lead to our first time system with "universal" in the name, Universal Time (UT). This was really just a disambiguation. Whether GMT started at noon or midnight required context. UT always started at midnight. And, since we're about to undergo a journey through universal times, we'll skip ahead and call it UT0. The IAU accepted this standard in 1884. Please note just how recent this is. In the history of human kind, precise time is a rather recent novelty. And note that people will always choose an epoch that is convenient to them. Even to this day, astronomers think of the day as starting at noon. (You'll see tables tabulated in Julan Dates(JD) that are all ending in 0.5, because JD is an astronomical concept built on noon, but all of the metrologists tabulated their data for days built on midnight)
Now time starts to get squirely. It turns out the planet doesn't spin straight. It has a wobble called polar motion. As a result, it turns out impossible to measure the duration of previous days using the current day as a "meter stick." So in the spirit of adding more mathematical corrections to make things more precise, in 1956 the International Time Beaudreau (BIH) adopted UT1 and UT2. UT1 included polar motion correction terms to once again describe time with respect to an abstract mathematical place where the "observations" were made. UT2 included seasonal variations, intended for civil consumption, but it wasn't as big of a deal because UTC soon came forth.
Coordinated Universal Time (UTC) arrived in 1963, along with International Atomic Time (TAI) at the onset of the atomic clock era. Atomic clocks had two huge advantages over Earth based measurements. First was that you could observe the TAI second using your own laboratory hardware, independent of the Earth's motion. The second was that as we got better duration measurements, we discovered the Earth's motion was astonishingly more complex than we initially thought. Started in 1955 (with the timescale named $T_m$), TAI measured time using the weighted average of atomic clocks.
This is a very important thing for your story. These clocks were not run in isolation. The standard was defined by a set of atomic clocks around the world which communicated using VLF radio signals. They worked together to keep the standard in sync, even though they were at a distance from one another. This quickly proved to be the most accurate time scale humans had created, and became the standard.
And here we have UTC. It got mentioned a few times on answers here. UTC is hilarious because, other than the name Coordinated Universal Time, its actually one of the worst possible "universal" systems to base a GTC around. UTC exists because we want the Earth to be privileged in some cases. We want one day to be one "Earth" day. And so, UTC has an astonishingly complicated history of slews and steps and updates, culminating in the leap second. If all goes well, the leap second will be abolished by 2035, although it will be replaced by an equally quirky leap-minute. And that is all I have to say about the compromise that is UTC here, other than to perhaps point out that even the acronym is compromise. It took several years of debate between English speaking countries, who wanted Coordinated Universal Time to be CUT, and French speaking countries who wanted Temps Universel Coordonné to be (TUC), to arrive at UTC which was equally wrong for both languages and started with U just like UT.
Let's get back to the interesting standards that apply to your problem. But first, I wish to revisit the problem of epochs. How does one pick an epoch? If one picks it arbitrarily, there's no way to convert between two systems. So typically we pick an epoch from the previous time system, and declare them equal. TAI, under the name $T_m$, was started with an epoch of 1958-01-09T00:00:00.0 which coincides with that date in UT2. There's no way to recover that date if it were lost (i.e. if all of the clocks shut down at the same time).
So how about them correction terms? In the 1970s, we noticed that all of the atomic clocks weren't operating at quite the same speed. They were at different altitudes, and thus experienced relatavistic differences. In 1977, TAI began accounting for this, making yet another abstract mathematical correction. It adjusted such that all clocks would be measured as if they were at mean sea level on the Earth. And this brings us to the start of solutions to your problems.
Solutions
At the time we began using relativistic corrections, we introduced three more time systems, Barycentric Coordinate Time (TCB), Geocentric Coordinate Time (TCG), and Terrestrial Time (TT). TCB, TCG, and TT are all relativistic corrected time systems. TCG measures time as it would be experienced by a point co-moving with the Earth, but not rotating and arbitrarily far from the Earth as to not be affected by our gravity well. TCB does the same thing, except instead of being tied to a point co-moving with the Earth, it co-moves with the barycenter (center of mass) of the solar system.
These show you the essential first step to your GTC. This is how our metrologists have defined a time system that has relativistic corrections put in place. Now per relativity, there is no truly definitive time system, but we can pick arbitrarily large systems and apply the same pattern. GTC would have a correction factor such that it measures time as an observer arbitrarily far from the galaxy but co-moving with its barycenter would experience. Using the barycenter like this does make a privileged point, but it has two advantages over other points. First off, it's fair. It really is the most democratic solution. The second appears when one starts to define reference systems to measure space. The barycenter has some unique properties that make it immune to really hard to work with relativistic effects like frame dragging. If you define time and space at the barycenter of the massive things you care about, the relativistic effects become manageable.
Note that all of these have been completely dependent on not only mathematical corrections, but on a consensus. BIPM asserts that no one measurement device can truly measure the current time in TAI, TCB, etc. It is defined as a consensus, after the fact. And, as such, the true "TAI time" of an event can only be known for events sufficiently far in the past as to have a published consensus about that time. You can approximate it really well, but it's not official until the official atomic clocks around the world have been compared and the metrologists have had time to analyze the details. The galaxy is 100,000 light years wide. This means any such consensus based time could only define exact time points for events 100,000 years ago. One would have to rely on a local time system and an approximate conversion. This isn't new. If you've ever set your watch to a site like time.gov, you've done this... just on a much smaller time scale. A question for you would be what time scale does your galactic society work on? Remember that they do need to coordinate to be a society, and that is unlikely to happen on any faster timescale, unless you add faster than light travel/communication to your world.
If you made it this far, and compared notes, you might notice I left one time system completely undefined TT. I left it to the end to highlight a fundamental limit of your GTC which cannot be overcome using current physics. TT is Terrestrial Time, and it realized via TAI: TT = TAI + 32.184s, as both TT and TAI are measured with respect to the Earth's Geoid. Why the 32.184s? History. TT was designed to read the same as its predecessor, Terrestrial Dynamic Time (TDT), at its epoch. In turn, TDT was based on Ephemeris Time (ET), a pre-atomic timescale designed to avoid the imprecision of the solar second of UT1. It defined the second as a fraction of the tropical year 1900 (356 * 24 * 60 * 60 seconds in a year). By the time TAI was defined w.r.t UT2, ET and UT2 had drifted by over 32.18 seconds. By the time ET was revised to use atomic seconds, the current difference of 32.184 was set in stone.
This shows something important for your GTC. There's no way to re-measure an epoch for a time system. They always daisy chain from one to the next. If you needed some "calendar event" to synchronize everyone in the galaxy, you will find it hard to come by. Better to have your GTC grow off of one civilization's most accurate time system, share an epoch, and then start ticking from that point on in a galactic way. Later, compare notes just like they do with TAI.
And do remember that this is relativistic time we're dealing with. I do not measure time the way you do. I may even see events occur in a different order than you do. However, by applying corrections, we can both determine what an abstract outside observer would see (comoving with the galactic barycenter), and we can both take the other's measurements and convert the back into our local time systems.