Thank you for putting this as a question. I'll try to answer.

GR is not restricted to describing our universe. It deals in general curved spacetimes, an infinite number of which are manifestly unphysical, a classic example being the Gödel solution of 1949. Physical applications of General Relativity require one to impose a number of constraints, most of which are motivated by evidence. I will restrict myself here to the constraint you asked about.

Our universe is up to measurability and for all practical purposes Lorentzian (3 spacelike and one timelike dimension), giving us local time-orientability. Our universe cannot be non-orientable for a variety of reasons, including the highly investigated parity symmetry breaking in the neutrino sector. The gaps in which one tries to hide from this are small and shrinking. This is a strong constraint, and it was felt when investigating SUGRA (especially the 11-dimensional maximal SUperGRAvity) as a physical theory.

The unit-timelike vector in the paper discussed in The Fine Article can, without constraints, break local time-orientability. One could grind out consequences of this, but it's not easy to do (and the authors do not do it in this paper). One might, for example, need a mechanism to suppress the popping-up of light right-handed neutrinos [1], or at least their impact on astronomical spectra and terrestrial studies of isotopes. I'd think one would want to do that in the matter side of the EFEs rather than the curvature side, but partly it's a matter of aesthetics and ease of use, and partly it's "you introduced curvature which has the effect of changing the observables predicted by the Standard Model, including regularly confirmed observables; you should repair the damage with curvature (and the obvious thing is to just hit 'undo')".

There are many other natural phenomena that serve as probes of local Lorentz invariance violation ("LIV"), which I am sure you can find if you are interested. https://en.wikipedia.org/wiki/Modern_searches_for_Lorentz_vi... is a decent jumping off point. One might note that there is a lot about neutrinos there.

GR can certainly completely consistently describe spacetimes of different dimension, different (or non-)orientability locally and globally, and so forth. There are good reasons to study 2-dimensional and 3-dimensional spacetimes instead of the 4-dimensional one we inhabit. There are fun reasons to study timeless, many-timed, and/or many-spaced spacetimes[1b]. There have been reasonably well-motivated-by-physics reasons to study extra spacelike dimensions.

I'm not sure what your threshold of "be[ing] sure" is, but mine is certainly met. It would be a tremendous (and cool) surprise if LIV were to be shown experimentally or implied in extreme astrophysical observations, and even more so within our (less extreme) solar system.

Finally, I turn to your IIRC, which is not correct. General Relativity has a single metric to which everything couples. This is the root of the universality of free fall part of the Strong Equivalence Principle. In General Relativity we can only vary the (vacuum) speed of light by giving light a rest mass. Most Variable Speed of Light (VSL) approaches introduce new objects into the field equations, making a theoretical alternative to General Relativity. Examples include bimetric theories, which add a metric of constant curvature and typically other scalar and/or vector fields; different massless species can couple to the different metrics, and if the constant curvature is Minkowski (flat), then Special Relativity is preserved on a per-sector basis, with complicated intersectoral interactions. There are a variety of other approaches, Magueijo 2003 https://arxiv.org/abs/astro-ph/0305457 is an overview that springs to mind.

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[1] Gordon M. McCabe's 2005 paper http://philsci-archive.pitt.edu/2218/ §2.1 (pdf page 14) expands upon this. (The author writes assiduously in this area with few, but not zero, citations https://scholar.google.com/scholar?hl=en&as_sdt=7,39&q=Gordo... and with occasional collaborations -- he is among other things one of the editors of https://ui.adsabs.harvard.edu/abs/2015rrrp.book.....S/abstra... https://en.wikipedia.org/wiki/The_Road_to_Reality )

[1b] ibid., §2.2 (pdf page 18) is the start of an extended discussion of how one might couple realistic matter (in this section a toy non-interacting (except gravitationally) field, but heading towards the full Standard Model in later sections) into non-Lorentzian universes.