What is the definitive evidence we have that quantum theory's probabilistic nature is physical and not epistemic?

I have no opinion about whether the wavefunction is "real" or "just a trick." I have never even understood what the debate is about, since you could have the same argument about any mathematical object (eg an electric field vector).

There is a genuine physics question, however, in asking whether quantum mechanics is a complete formalism, or if there are additional "hidden variables" we don't observe. Perhaps our statistical interpretation of quantum mechanics is simply a coarse way of describing the behavior of these hidden variables, but if we could only find them we would see that physics really was deterministic after all.

Then the typical response is to point out that experimental demonstration of Bell's inequalities rule out that quantum mechanics can be explained by a theory where the "hidden variables" interact locally in a deterministic way.

At this stage, you can either accept that quantum mechanics is the simplest and best explanation of the data, or you can push into increasingly tiny loopholes in the experimental tests to find room for something other than quantum mechanics to explain what we have observed. There is a huge literature that has a cat-and-mouse type feel, with loopholes being proposed, and experiments closing the loopholes.

On your first question - whether wavefunctions and superposition are 'real' or 'a model making accurate predictions' - there is no way to tell. This is metaphysics. Since by definition there is no observable difference between true and false models so long as they both make accurate predictions, there is no scientific way to distinguish them.

On your second question - whether the probabilistic nature of the Copenhagen interpretation (wavefunction collapse to an eigenstate of the observation) could be only 'apparent' - the answer is 'yes'. The Everett interpretation (also misleadingly known as the 'many-worlds' interpretation) is deterministic, local, complete, time-reversible, and realist. There is no randomness, no spooky faster-than-light action-at-a-distance, no unexplained irreversible 'wavefunction collapse' process triggered by some special vaguely defined property of observation.

Quantum superpositions and wavefunctions apply at the sub-atomic level, and both sides of the debate accept that. The Everett interpretation adds no new physics to this, it simply points out that the quantum theory predicts that we will see at the macroscopic level exactly what we do see - i.e. a quantum universe appears classical to quantum observers inside it.

The principle by which this works can also be seen in classical physics, in the case of 'normal modes of vibration' of coupled oscillators. Here, two independent oscillators can be jointly represented with a matrix equation $mathbf{ddot{x}}=Mmathbf{x}$ where the matrix $M$ is block-diagonal. (Compare with the Klein-Gordon equation in quantum mechanics.) When a weak coupling is introduced between the oscillators, by setting some of the off-diagonal terms non-zero, the two oscillators enter a superposition of synchronised states, which are related to the eigenvectors of the modified matrix. The eigenvectors are orthogonal to one another, meaning they oscillate independently. This is how the Everett interpretation says an observation works. The two independent oscillators are the observer and the observed. The weak interaction between them is the process of observation. The synchronisation of oscillations in each normal mode is the observer state becoming correlated with the observed state (i.e. acquiring information about it). The orthogonality of the normal modes is why each observer state can perceive none of the others. If the observer is a quantum oscillator, weakly interacting, then it will enter a superposition of states, each of them corresponding to only one possible outcome, each of which behaves completely independently of the others, as if none of the others exists. It is 'as if there were separate universes, in each of which one of the possible outcomes occurs, all existing simultaneously' (hence the name 'Many Worlds').

Everett's original thesis is well worth reading. There are lots of misunderstandings and distorted versions floating around. There remain some genuine philosophical problems about it, but the same is true of all the others, and it is deterministic, local, complete, time-reversible, and realistic, which should all count in its favour. And in this interpretation, the probabilistic nature of quantum mechanics is only apparent, so it is at least possible.