Can there be a theoretical synchronised ‘now’ moment at all points across the universe?

It does not make sense to "stop time in an instant across the universe" because time does not stop. You can choose an instant across the universe on a spacelike hypersurface, but this is assuming the conclusion, since your spacelike hypersurface (your now moment) must be chosen in order to choose an instant. Moreover, there are an infinite number of ways of doing that.

However, we can and do define Cosmic time, which defines a spacelike hypersurface in a fairly natural way. Cosmic time does not define an absolute concept of time (analogous to Newtonian time) but is a way of synchronising the independent times defined locally by clocks. One way to define it was suggested by Weyl. I have described this in Structures of the Sky

Weyl noted that we define synchronous slices of space in our own neighbourhood of the universe, and assumed that it is possible to extend this definition by defining a synchronous slice in a region centred on a point at the edge of our neighbourhood. An observer can use the radar method to define synchronous surfaces in his neighbourhood with respect to his own proper time, as we do when we define time in the Earth frame (figure 13.3, top left). Earth time may be synchronised to solar time, measured according to the Earth’s orbit and applicable to planetary orbits (figure 13.3, top right). We may further synchronise to galactic time (figure 13.3, bottom left), where the radar method is not viable, but in which the time of an event can be estimated from the distance travelled by light. Cosmic time assumes that it is meaningful to think of a time parameter synchronised across galaxies and groups of galaxies (figure 13.3, bottom right).

Ignoring peculiar velocities (which are small compared to the speed of light) Cosmic time enables us to describes the age of the universe as being the same for each galaxy on a sychronous slice. It is the time coordinate used in the Friedman solutions.

This is not really an answer. It’s more of a continuation of your thought experiment. Maybe it will help to think of the question in different ways.

Imagine that every particle in the universe had its own personal timer that said “exactly 20 billion years after the big bang, I need to record what I’m doing at that instant. Then when someone asks me what I was doing at that instant, I can tell them.”

I’ll refer to this as the “task”. I can see several problems with this task.

problem 1: Photons themselves do not experience the flow of time. Photons travel at the speed of light and from their point of view, time does not pass at all while they move from point A to point B. So right off the bat I have to exclude photons from this task. I can’t think of any way to include them.

problem 2: Suppose that massive particles could perform this task. This task still makes little sense for individual particles. Maybe it could make slightly more sense for large aggregations of particles. So if I was able to, say, tell all the particles on earth to average their clocks together and use that instead, then maybe that would make more sense.

problem 3: Of course, the act of telling all the clocks on earth to average themselves together and synchronize themselves would take about 1/10th of a second to get that message across. Another can of worms.

Problem 4: Then you have the problem of a satellite orbiting the earth. Time passes slower on a satellite orbiting earth. 0.01 seconds slower every year. So over the course of 20 billion years, that works out to 6.32 years difference between a clock on earth relative to the space station. Hmm.

https://www.wolframalpha.com/input/?i=20+billion++Years+*+0.01+seconds+%2F+1+year

Problem 5: Once an I record my state at my own 20 billion year mark, then I have to begin the process of going around collecting all that data from everyone else.

You can definitely choose a 3 dimensional surface in spacetime that has some properties of a "now" surface. For example, you may ask:

• The surface is spacelike. This means that all tangent vectors to the surface are spacelike. We would like to include this condition. Otherwise, "now" would contain two events which could be reached by a particle at different instances of time, when "now" should only contain one instance of time.

A strengthening notion is that of a Cauchy surface. This is a surface where every inextendible timelike curve crosses the surface once. This is the type of surface where one would put initial conditions for a field on general relativity. Essentially, all of the dynamics of such a field would be determined by its initial configuration and derivative at our "now", much like Newton's second law determines the evolution of a particle given its position and velocity at an instant of time. Not every spacetime admits this kind of surface. One that does is said to be globally hyperbolic.

Of course, one looses things by identifying the above as "now" in the sense of previous theories to GR. For example, in general there is no observer who will identify that surface as a now. In general, observers can only give a sense of now locally. Secondly, the structure of now may change from time to time. For example, one can have a perfectly smooth "now" at some instance only to find that the "now" at a later time has a singularity (say, a black hole). In general, there is no way of identifying the "now" at one instance with the "now" at another. This is what makes canonical (as in Hamiltonian) formulations of gravity so difficult.

It underlines how the notion of ‘now’ is compromised due to time passing at differing rates in differing frames of reference

I think your confusion stems mainly from this. The fact is, that logic is reversed to what you are saying. You cannot compare two clocks before you know what it means for two events to be simultaneous. To say that two clocks tick at different rate means you need to start and finish counting the ticks at the same time for both clocks and then you compare the resulting number. So you cannot even compare two clocks before knowing what "now" means.

Relativity was born from Einstein's careful analysis of what "now" should mean. The definition he came up with led to the fact, that "now" is different for different observers. From this, STR follows.

if one could stop time in an instance across the universe, would that not represent a common now moment?

You can do this, but this would not represent common now moment. This would be now moment from your viewpoint. From someone else's viewpoint it would not. You can imagine that you send signal with infinite velocity (w.r.t to you) to whole universe which will tell him to stop. But from other viewpoint, this signal would not have infinite velocity, but finite and this observer would see universe stop in different places at different time. For him, this would not represent "now" moment.

if one could stop time in an instance across the universe, would that not represent a common now moment?

Your condition of "an instance" presupposes a common now moment. One might equally stop each point in space at an arbitrarily different moment, who is to say which moments might be synchronised and which not? In other words, your question is a bit like, "Given that Relativistic time is not Universal, would Universal time be a good model for it?" Of course, if it were true that one could identify a Universal "now" then it would be true, but Einstein showed that it is not.

The best we can do is to note that each observer sees only their own light cone. Outside of their own light cone there is no defined past, present or future. Despite all the apparent warping of everybody's light cones via gravity and relative motion, when you overlay them all they always give the same consistent sequence of events along any given timeline. They just disagree how far apart the events are.

You can think of spacetime as a four-dimensional "block" Universe. Over that, each observer lays a reference grid a bit like the lines of latitude and longitude on a map of the Earth, together with altitude and time. Each such grid has its "now" moment or slice, but exists only within the observer's time cone. Take multiple relativistic observers and overlay all their grids on the block Universe, and the grids all overlap and clash with each other. The "now" slices are all hopelessly criss-crossed. There is no way to arrange them Universally so that every observer will agree.

The effect is predicted by Special Relativity. It remains even when one has corrected for the General Relativistic effects of gravity and acceleration. One observer can still calculate that events A and B were simultaneous, while another calculates that, after corrections, A happened first and another, after similar corrections, that B was first. It arises as a consequence of the constancy of the speed of light for all observers. The exact explanation for that is some mildly complicated maths which you will find in any text book on Special Relativity.