I am not an expert in GR, so I have the same kinds of questions, but I'll throw out what my answer would be in hopes that real experts will be able to correct any misconceptions. It seems to me classical mechanics proceeds in essentially (at least) three separate stages. The most elementary stage we might call "particle dynamics", which is essentially Newton's laws, which describe how the fundamental "atoms" (classically) of a system behave (i.e., accelerate) from the point of view of observers instantaneously and inertially moving with these fundamental dynamical elements. As different elements have different motions, it is thus necessary to "cobble together" the information obtained by all these hypothetical comoving observers into the "local dynamics" of a confined closed system, and the instructions for how to properly combine the information from all these comoving hypothetical observers into a view of one observer generates a "tangent space" where the local dynamics plays out from the perspective of anyone among this set of local observers, and the instructions are mediated soley by relative velocity (one can imagine noninertial observers too but I don't think that adds anything locally). However, tidal gravity effects warp the relative actions of neighboring tangent spaces, such that integration of nonlocal information into a more global whole requires additional instructions about how to treat gravity, and that's what general relativity supplies. So in summary, Newton's mechanics gives us the proper accelerations, special relativity allows us to track the local interrelations of these accelerations, and general relativity allows us to piece the local information into a global description including gravity.
Thus it doesn't matter how we coordinatize the tangent spaces, because GR will take that into account in the connecting instructions. But if we choose coordinates of each tangent space in a way that transforms under Lorentz transformations to different observers at the origin of the tangent space, then we can at least handle the "cobbling together" of each tangent space automatically by following simple covariant instructions for how to piece together Newton's laws, and will only need GR to handle tidal effects (second order effects) as we go between neighboring spaces.
So in other words, there are two different things going on here, the coordinatization, which is purely arbitrary and just involves giving labels to everything, and the physics, which takes into account the labels to determine how the information compiled by individual observers can be put together into a whole. That compilation proceeds in three stages, the particle dynamics, which are just Newton's laws for comoving inertial observers, the local dynamics, which require special relativity (which can be made transparent by choosing a covariant labeling system), and the global dynamics, which require general relativity because no globally covariant coordinatization exists to make the global dynamics transparent.
Thus my answer to "is there anything physical in a coordinate transformation" depends on whether one thinks of coordinates as being ways to achieve transparent dynamics, or if one is willing to do considerable work finding the dynamics after the coordinates have been chosen. It seems to me the point of a sensible coordinatization is to minimize that remaining work, so if that is the philosophy used, then there is something physical, not in the coordinates themselves, but in the way they get chosen. That holds both in the tangent space (Lorentz covariance) and in the global connections (which might, for example, respect the cosmological principle).
