tl;dr:

> I'm genuinely confused about what we're disagreeing on.

That our universe is locally Lorentz-invariant everywhere.

> I think this is a perfectly reasonable interpretation of my original claim that Bohmian mechanics doesn't have a field theory.

I don't, however charitably I am willing to read this as "Bohmian mechanics doesn't have a relativistic field theory" given the new material in your latest comment. "Relativistic" means that there is a Lorentzian manifold involved, either directly or as emergent property of the configuration space.

> front and center

No, what puts Lorentz-invariance front and centre in this context is the Standard Model Extension <https://en.wikipedia.org/wiki/Standard-Model_Extension> which can carry the parameters for a de Broglie-Bohm pseudo-relativistic field theory which has additional Lorentz-invariance-violating (LIV) structure. Tests of such parameterized theories have all failed unless the controlling coefficients are designed to vanish beneath the floor of current-technology testability (in which case is the more complicated theory useful?).

There is no evidence at all for LIV physics in direct tests in multiple locations in our solar system, in observational tests within our solar system, or beyond it. Lorentz-invariance is probably the strongest-tested aspect of modern physics, as it forms part of any test of the standard model of particle physics, quantum electrodynamics, or general relativity. <https://en.wikipedia.org/wiki/Modern_searches_for_Lorentz_vi...>

> irreconcilable differences between relativity and QM

There are no such differences between special relativity and relativistic QM, nor can there be because of the design of the latter type of theory. So you must be arguing that quantum mechanics is not special-relativistic. OK, more on that below.

The only thing that general relativity brings to the table is that global Lorentz-invariance becomes local Lorentz-invariance ("local" can be an arbitrarily large finite region, however) becomes ultralocal Lorentz-invariance in strong curvature, as in the radius of curvature is much less than the characteristic lengths in quantum chromodynamics (e.g., the available free-fall trajectories lead to spaghettification which ought to produce hadronization of the tidally-disrupted QGP or colour string, where the resulting hadrons are then themselves subject to tidal disruption; for a slowly-spinning black hole we're interested in the axial stress as the Weyl tensor deforms sphere->lengthening American football, one end pointing in, one pointing out->lengthening thin tube with slightly fatter middle than ends->...).

Since in GR strong curvature is hidden within various types of horizons there is no need to worry about proving ultralocal Lorentz-invariance, rather than proving e.g. the existence and behaviour of horizons (which are more readily testable features of GR). Maybe someday we'll watch a black hole fully under our control evaporate completely; or be able to induce extreme angular momentum and/or electric charge on a black hole we control.
Until then, all we can do is speculate that either Lorentz-invariance holds up even in black holes and during the hot big bang, or the LIV in those regions washes out of our present-day observability through some unknown process.

> if it can be overcome

Well, right, you seem to be claiming that the universe is not locally Lorentz-invariant at every point. That is, that the universe is not relativistic. Fine, that's a claim which has been made before, and it is almost always a physically testable claim. Now try to prove it, like the other claimants. It may take considerable effort to do so in some "ontologically pure" way, but that's on you and anyone you convince to collaborate with you in this programme of research.

I promise that any serious effort would be taken seriously by at least some relativists. If nothing else, they would show you how to cast your attempted proof into a more standard test theory, and in particular how the test theory's coefficients relate to your proposal, and how they can be compared to experiment and observation.

(Clever people like Clifford Will do this in gravitation with some regularity <https://en.wikipedia.org/wiki/Alternatives_to_general_relati...>.
There are numerous parametrized test theories of a range of aspects of the Standard Model and BSM particle physics, too.)