The correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers. This is due to Born rule working or are they independent?
We really have three questions:
Why is there an inner product (which provides a measure)?
Why is there a probability interpretation?
Given that there is a probability interpretation, why does it have to be given by the measure?
1 is pretty easy to at least construct a plausibility argument for. Quantum mechanics isn't really a physical theory if you omit the inner product. If you don't have an inner product, then you can't define whether an operator is normal. This means you can't define which operators are valid observables.
Without a measure, there will also be difficulties tying QM to the background of space. In the position basis, you can't compare the amplitude or phase of a wavefunction at different points; you can't define differentiation; and you can't define a momentum operator. You can't define what it would mean for a wave packet's envelope to slide along, so you probably can't define galilean invariance or the notion that a hydrogen atom here is the same as a hydrogen atom there. You can't define the Ehrenfest theorem, which along with the inability to recognize valid observables probably means no possibility of making any contact with a classical limit.
3 is fairly straightforward as well. People have formalized it, but there are also easy informal arguments.
As an example of an informal argument, suppose we do double-slit interference with photons. If we run the double-slit experiment for a long enough time, the pattern of dots fills in and becomes very smooth as would have been expected in classical physics. To preserve the correspondence principle, the amount of energy deposited in a given region of the picture over the long run must be proportional to the square of the wave's amplitude. The amount of energy deposited in a certain area depends on the number of photons picked up, which is proportional to the probability of finding any given photon there. This doesn't quite work as a formal proof, because photons don't actually have a wavefunction $\Psi(x)$. For a more careful and rigorous proof, see Gleason's theorem.
The real question is 2. Why should we experience the universe in such a way that when we look back over our experience, we seem to see patterns of behavior that can be summarized by statistical rules? I haven't seen a satisfactory ab initio explanation of this from any more fundamental principles, and if you were looking for such an explanation, I wouldn't know what more fundamental principles to start from.
I think discussions of this sort of thing tend to veer off into religious discussions of MWI versus the Copenhagen interpretation, but IMO that's an orthogonal issue. Neither interpretation pretends to answer #2, and neither interpretation is necessary in order to answer #1 or #3.