Still no violation of Lorentz symmetry, despite strongest test yet

2 very sensitive measurements conducted within the past year seem to suggest (if GR is true), that we are in a universe that lacks causality. 2/3 LIGO detections imply one of the merging pair of black holes should be a naked singularity.

GR allows naked singularities. It models them fine. GR just stops being globally deterministic.

If you look up the history of GR some mathematicians in 50's made some really weird proposals for non-causal universes that would appear locally causal. But there isn't a way to test this. So it is more pure mathematics or philosophy then physics.

For example, a minimal SME photon Lagrangian is developed in section II. A. of Kostelcky and Mewes @ https://arxiv.org/abs/0905.0031

The difference between a minimal SME (mSME) and the full SME is that the latter admits terms of any mass dimension greater than two, while mSME requires power-counting renormalizability (in order to be an effective field theory) so mSME admits only terms of dimension four. As a result mSME can only have a finite number of Lorentz invariance violation parameters, while SME can have an infinite number of them.

The bright side is that mSME can represent locally the Lagrangian in the physics.se answer you link to with a fixing of parameters; the dim side is that in general one does not want to do this a priori, precisely because the mSME is a tool to investigate whether the Lorentz SO(3,1; R) group is an exact feature of nature at every point, rather than postulating that it is (as the Standard Model does, and indeed as Special Relativity and General Relativity do).

For example, SME is useful for dealing with things like this: https://www.wikiwand.com/en/Bumblebee_models#/Lagrangian where the non-matter term might not reduce (via GL(4, R)) to the groupoid of local coordinate transforms on the manifold and the local SO(3, 1; R) Lorentz group on the orthonormal frames (tetrad formalism) which are the symmetries of General Relativity. The point is to find out under what conditions any such model does vary from General Relativity. (We already have a good EFT for GR itself, and the result of the Bourgoin et al. paper is that it remains good.)

In SME you write down an observer Lagrangian incorporating tensors, spinors, covariant derivatives, and so forth, and SME coefficients: L = L_gravity + L_SM + L_LIV + ... where SM is the Standard Model.

You can write down an L_LIV that is an extension of GR with or without torsion. The second you introduce nonvanishing torsion you are no longer in Riemann spacetime. Without torsion, however, you can keep yourself in a Lorentzian manifold and still introduce e.g. spontaneous (and thus preserving geometrical identities) Lorentz breaking in the L_LIV, as in Bumblebee models.

In General Relativity, Lorentz invariance is what you get when you put a metric with a signature of (-+++) or (+---) into the Einstein Field Equations and then look at the local isometries induced on the tangent spaces and find the O(3,1) group. But you could just as easily plug any signature into the EFEs - there's no mathematical restriction, and there are researchers who use all sorts of signatures (e.g. (+++++), which is manifestly not Lorentz invariant).