The way I view things is that the Lorentz transform is a transformation matrix, not a tensor.
We use tensor notation to write its components, just as we use tensor notation to write the componetns of Christoffel symbols. But neither is a tensor. The fact that the Lorentz transformation doesn't directly apply in Rindler coordinates is a practical illustration of the non-tensorial nature of the Lorentz transform.
Because the Lorentz transform isn't a tensor, one needs detailed assumptions about the coordinate system that's being used to know when it applies. Tensor expressions work in all coordinate systems, non-tensor expressions are not so general. There are various possible ways of specifying coordinates. Usually in physics, specifying a line element is regarded as the simplest way of communicating the choice of coordinates. See for instance Misner's "Precis of General Relativity"
https://arxiv.org/abs/gr-qc/9508043
misner said:
Equation (1) [[note: this is the aforementioned line elmenet]] defines not only the gravitational field that is assumed, but also the coordinate system in which it is presented. There is no other source of information about the coordinates apart from the expression for the met-ric. It is also not possible to define the coordinate system unambiguously in any way that does not require a unique expression for the metric. In most cases where the coordinates are chosen for computational convenience, the expression for the metric is the most efficient way to communicate clearly the choice of coordinates that is being made.
So from this point of view, when one choose one's coordinates in any space-time, one necessity define a line element that describes the geometry of that space-time by defining the invariant interval between nearby points in that space-time.
Note that in general, over large distances, the invariant interval between points along a geodesic connecting said two points will not necessarily be a quadratic form. Consider for instance the full expression for the distance between two points on a sphere for example. Furthermore, there won't necessarily be a unique goedesic between two points in a general space-time. However, if the points are close enough, there's always a unique geodesic between them, due to the existence of local convex neighborhoods in topology, and also some function that does give the interval along this unique geodesic. The low order series expansion of this function turns out to be a quadratic form in the differences between coordinates, which is the line element that defines the geometry.
When this line element is in the Minkowskii form ##dx^2 + dy^2 + dz^2 - c^2\,dt^2## the Lorentz transform applies. When it's not in that form, the Lorentz transform doesn't necessarilly apply.
Globally, when coordinates exist such that the line element is Minklwskii at all points, we have the flat space-time of special relativity, and the associated coordinates define by this line element (recall the previous remarks I cited from Misner on this point) define an inertial frames of reference.
While the Lorentz transformation is not a tensor, local Lorentz invariance is a coordiante-independent notion. If one considers a single point in space-time, there are multiple choices for an orthonormal set of basis vectors at that point. Smooth transformations between these choices of basis vectors exist, and these transformations will be members of the Lorentz group (which includes rotations as well a Lorentz boosts, and possibly reflections as well - I'd have to refresh my memory on the last point).