>"Quantum Mechanics and General Relativity, contradict each other so at least one, and probably both, are approximations to the real laws that govern our universe.
[...]
overwhelmingly likely prior that General Relativity is correct"
If you think both QM and GR are likely incorrect, then why do you use "a overwhelmingly likely prior" that GR is correct?
Neither is "incorrect"; the Standard Model and General Relativity are two of our best physical theories in that they both accord entirely with observational and experimental evidence to date.
Either or both may be incomplete, however. Correctness and completeness of any theory in mathematical physics are esentially orthogonal. You can have a complete theory that is just wrong, for example.
As I wrote a bit earlier in this thread, the most straightforward approach to quantizing General Relativity fails in strong gravity. Additionally, the classical field theory that is General Relativity is defined on a smooth manifold and yet so far we have been unable to escape the conclusion that some systems of mass-energy inevitably produce a non-smooth discontinuity. A completion of classical General Relativity requires the smoothing of these regions. Sharpening this, the problem with GR is the prediction of a gravitational singularity; if singularities are physical at all (even if they are in a region of spacetime that is inaccessible outside event horizons), then General Relativity is incomplete in its own terms.
The Standard Model as a paradigm of quantum field theory, on the other hand, is defined against a flat spacetime and thus relies on the result from General Relativity that the flat spacetime metric is induced on the tangent space of every point in a smooth spacetime. So if GR is incomplete, so is the Standard Model, in its own terms. (This is not just an academic point; any theory of gravity that does not reproduce the Poincaré invariance of flat spacetime in the energy scales of the Standard Model has a terrible correctness problem.) Additionally, the Standard Model is not especially well-defined at GUT energy scales. Additionally, the Standard Model does not describe the whole of the non-gravitational content of the universe; for example, it is silent on dark matter.
The Standard Model is highly correct, however, in the limits where it is effectively complete. It's a pity it has so many free parameters that have to be determined by experiment.
Likewise, General Relativity is both highly correct in the limits of present observability, and it is complete in its own terms if one admits the possibility that gravitational singularities only arise in our idealized models and that, for example, there are no exactly Schwarzschild black holes anywhere in the past, present or future of our universe. (One have to show that, and also that there are no other physically realizable systems of matter that can generate non-smoothness in our spacetime. That's not an easy ask. Although General Relativity has only one of the free paramaters complained about in the previous paragraph, it doesn't offer much guidance about how to show that you can't actually generate a low-Q Kerr-Newman metric in reality, and worse, some of that guidance must come from the high-energy behaviour of matter fields -- we can only be as complete as the Standard Model right now.)
If you think both QM and GR are likely incorrect, then why do you use "a overwhelmingly likely prior" that GR is correct?