The insertion would not change the truth value, I suppose. But you apparently don't understand what is actually required for a counterexample. A counterexample would look like this: "Here's an actual physical observable that the standard EFE/SET method doesn't predict or explain." Or: "Here's a prediction made by the standard EFE/SET method that doesn't match this actual physical observable." You have given no such example, because you have never actually tried to figure out what the standard EFE/SET method predicts or explains; you haven't used it. You've insisted on reasoning from your own set of premises (like "gravity gravitates") instead, and then you've tried to claim that if the conclusions you reach don't appear to be consistent with the standard EFE/SET method, the standard method must be wrong. So it's not that I'm saying any counterexample must be wrong by definition: I'm saying you have not actually given counterexamples at all; instead you've given conclusions derived from a different set of premises altogether, and those premises are only approximately true (and even that is only in a limited domain).
Neutron stars are a good example of pressure contributing significantly to the SET, yes. And yes, the maximum NS mass is one area where the pressure contribution is important; we know that even though we don't know the exact equation of state (because we've tested a whole range of possible equations of state numerically).
I appreciate that things look this way from your pov. But now consider how they look from my pov. As I've said several times now, in the standard EFE/SET picture, there is no *need* for the concept of "gravitational energy" at all. All physical predictions can be made without ever using it. So from my pov, the problem is not that I'm not answering your questions, but that you insist on asking them even though I've repeatedly said that they are based on the wrong set of concepts. I have been trying to meet you halfway by at least trying to express how one *might* salvage some kind of correspondence between the concept of "gravitational energy" and the standard EFE/SET method, in a limited domain. But that's only because I understand that the concept of "gravitational energy" has intuitive force, so I'm willing to expend some effort in trying to explore it and its limits.
But asking for what "exactly" the concept of "gravitational energy" means is asking too much: the concept is only a heuristic one and it does not have an "exact" meaning. (Or perhaps a better way to say this would be: one could give an exact definition of "gravitational energy", such as the Landau-Lifshitz pseudotensor, but no such definition is unique, and any such definition only "makes sense", only corresponds to our intuition, in a restricted set of cases.) If you want an exact answer, it is this: there is no "gravitational energy" in the SET, so as far as exact calculations of physical predictions are concerned, it doesn't exist. (You'll note, in this connection, that nobody uses any definition of "gravitational energy" to actually make physical predictions: they all use the standard EFE/SET method, and then once they know what the answer is, they overlay their chosen concept of "gravitational energy" on top of it to help them understand intuitively what's going on.)
All right, let's look at this from an *exact* point of view. The exact point of view is this: the "total system" is the entire spacetime, including the region "at infinity". This "total system" does not *have* a "mass M". The exact metric is not in any of the forms where "M" even appears; it's more complicated. (One could try to extract a "piece" of the metric where a coefficient "M" appears, but that's just an approximation-see below.) So from the "exact" point of view, there is *nothing* in the physics corresponding to "total system observed mass". There is a metric at each event, and there is an SET at each event (nonzero in the interiors of the two pulsars themselves, zero everywhere else--if we ignore the EM radiation emitted by the pulsars and assume the only "radiation" in the spacetime is GWs), and the EFE holds at each event. That's it.
Does this "total system" have a "total energy"? It depends on how you define "energy". The spacetime as a whole does not have a time translation symmetry, so we can't define "energy" that way. The spacetime *may* have a continuous set of spacelike slices that match up well enough with what symmetry does exist (for example, maybe the slices are good approximations to "natural" ones that observers hovering at a large radius R above the binary pulsar system would pick out as "surfaces of constant time") to be useful in defining "energy" by integrating the energy conservation equation (i.e., the covariant divergence of the SET) over each spacelike slice. This could define a "total energy" for the system, and this total energy could turn out to be conserved (i.e., the same on every slice), at least to a good enough approximation (the same level of approximation to which the slices are good "surfaces of constant time" for some set of observers). But will this "conservation of energy" be "exact"? Probably not, since the spacetime does not have any exact symmetry. So if you want an exact answer, it is that there is no "total energy".
Now, suppose I decide to draw a boundary at some finite radius R around the binary pulsar system, and say that inside that boundary is "the total system" and outside it is "the rest of the universe". I can pick R large enough that, to a good approximation, the binary pulsar system "looks like" a simple gravitating body with some mass M. More precisely: the metric at R is still not quite in the Schwarzschild form, because the spacetime is not spherically symmetric or static; but it will be close enough that I can "split" it, approximately, into two pieces: a "Schwarzschild" piece and a "gravitational radiation" piece. The Schwarzschild piece, to a good approximation, will look like a gravitating body with a mass M that slowly decreases with time ("time" meaning "proper time according to an observer hovering at radius R). The gravitational radiation piece will be oscillating in quadrupole fashion, and could be measured by, for example, letting the oscillations heat up a detector and measuring the energy taken up. We could then, in principle, do an energy balance: the decrease in M is balanced by the energy carried away by GWs.
Will this energy balance be "exact"? Probably not, because the split of the metric into the two pieces probably won't be exact; there will probably be extra terms in the metric that are left out--they aren't included in either the Schwarzschild or the GW piece--because they are small compared to both of those pieces.
So we come back again to what I said above: if you insist on an "exact" answer, then it is this: "gravitational energy" doesn't exist, and the only exact "energy conservation" is what I said earlier: the covariant divergence of the SET (the standard SET) is zero at every event. Anything else is approximate, and breaks down if you try to press it too hard. That includes things I've said previously (like "M cannot decline if all matter-energy is included"); I apologize if I didn't make it clear enough that I was only speaking approximately.