Unfortunately, if the "new theory" actually "gets away" from this -- which I seriously doubt it does -- it cannot possibly be correct. Belle's Theorem is not an argument: it is a rigorously correct mathematical result. The Einstein-Podolsky-Rosen Paradox is not a paradox at all: it is a description of what actually happens in our world. Claiming that the correlated spin of one particle doesn't really effect the other at a distance because this is simply "the reality" and the other possible correlated spin state is "not a reality" is tautological.
To take a simpler example: The behavior of sufficiently smooth functions on their boundaries completely determines their behavior throughout the volumes in which they are defined. This is a (admittedly imprecise statement of a) mathematical fact. Consequently we can describe our physics as happening throughout the volume of space-time, or, if we desire, we can consider that reality an illusion for something that is happening simply on the boundary of the universe with the same physical laws (suitably mathematically rephrased on that boundary). This has led to some silly articles about how we may be nothing more than "holographic projections of something that is happening on the edge of our universe." Well, actually, we are, and we are also happening inside the physical volume of the universe, because the two things are not different. The physics is not changed.
Back to the instant case: the fact that the cat lives sometimes and dies other times is not altered by this new interpretation. Unless the "new theory" can actually predict when the cat dies, the uncertainty principle, non-deterministic causality, and all the so-called "paradoxes" remain, and are not answered by hand waving arguments that in some cases the cat lives because that's the way it goes, and in some cases she dies because of the same "explanation."
Sorry, but it has the stench of the weak anthropic principle: either it's trivially true, or it's not science. I hope there is a lot more to this new "theory" than the article conveys.
I think you are getting close to understanding what they are trying to get at, but rejecting the point by referencing the weakness of standard Quantum Mechanics to deal with this question as the reason this solution can't work when the contrary is exactly the point.
This Invariant Set Postulate concept is that QM is ignorant of the existence of the "reality" that is the Invariant Set and thus predicts possibilities that are within this Set and those which are outside of this Set, but only those which are within the set are manifest in "reality"
To go to your basic point about smooth functions and a smooth predictable volume within the Universe, this idea sees this volume as a much more complex space wherein many functions may have described the Volume but only a select few indeed do. The insight is that the Quantum Mechanical states possible are restricted to a specific set of states that are only a small group of the states that would otherwise be possible and this set of real possibilities is what we term as physical reality.
What I see missing from this discussion is the fact that the "Invariant Set" must by force be collapsing and thus I am unsure about the name Invariant which would imply that all events throughout all time are already pre-ordained and though potentially not predictable are yet inevitable.
My best guess is that this is not the meaning Palmer has meant to project by using the term Invariant, but that is the limit of this description.