Putting aside the specific definition of "Mach's principle" (which seems to mean different things to different people), as well as sociological questions about what relativists usually do or don't do, the way I understand it is as follows (and I would welcome corrections from anyone in a position to offer them):

In general relativity, spacetime is a Lorentzian manifold -- that is, a manifold modeled on Minkowski space, which is just like Euclidean space except with a funny inner product such that the "norm" is constant on hyperboloids instead of spheres -- which is itself determined, according to a system of hyperbolic PDEs known as the Einstein equations, by a particular tensor field (representing "matter") on it. (It's not clear to me whether what is being "determined" here is just the Lorentzian structure, or if one somehow "solves for" the underlying topological and differentiable structures of the manifold as well; but leave this point aside for now.)

A "frame of reference" is actually the same thing as a "state of motion": they are both physicists' jargon for a chart of the manifold, i.e. a mapping that serves to identify a particular open set (or "locality") of the manifold with a corresponding open set in the model space, which in our case is Minkowski space.

Part of what it means to be a manifold of a given type (e.g. topological, differentiable, Lorentzian) is that any two charts defined on the same locality are considered equivalent, so long as the resulting mapping between the two corresponding open sets in the model space (called a "transition map") preserves the structure in question (topological, differentiable, Lorentzian, etc). In our case, then, any two charts defined on a locality are equivalent, provided that the transition map is "Lorentzian" -- which, one has to assume, must mean that the derivative of the transition map, at each point, is a Lorentz transformation (a linear operator on Minkowski space which preserves the "Minkowski norm" defined by the funny inner product mentioned earlier).

In other words, every "change of coordinates" (transition map) which is locally a Lorentz transformation is to be considered "legal". When physicists say that there is "no privileged frame of reference", they are not saying anything not already contained in the statement that spacetime is represented by a Lorentzian manifold. (Of course, it is not true that there is no privileged class of reference frames: to be a member of the privileged class, a chart must be "Lorentz-compatible" with all the other charts in the privileged class.)

Now, what about curvature? Well, note that in order to even be able to say that a Lorentzian manifold is capable of having curvature, the notion of "curvature" must be chart-independent -- otherwise, it's only the charts that have "curvature", and not the manifold itself. It turns out -- so mathematical legend has it -- that there is such a "global" notion of curvature that makes sense for Lorentzian manifolds; and in fact it is (more or less) the very thing that the Einstein equations say is determined by the "matter" fields on the manifold.

(Roughly speaking, what it means for the spacetime manifold to be "curved" is that you can end up "heading in a different direction" after a while despite being "at rest" the whole time.)

So, the point is that there is no way a "legal" change of coordinates can ever change the curvature of spacetime. This isn't a matter of the convention of physicists, except insofar as they have decided to represent spacetime as a Lorentzian manifold rather than some other type of manifold (or some other kind of mathematical structure altogether). If you want to be able to change the curvature, you have to allow non-Lorentz local coordinate changes, and thus work in some other kind of mathematical structure different from a Lorentzian manifold.

(And I don't think that's what Eliezer was suggesting.)