The answer to this question is that active diffeomorphism invariance is maintained but only for (low-energy) internal observers, i.e., those observers who can only perform (low-energy) experiments involving the propagation of the relativistic collective fields. By revisiting classic Lorentz–FitzGerald ideas on length contraction, and analyzing the Michelson–Morley experiment in this context, it has been explicitly shown in  that (low-energy) Lorentz invariance is not broken, i.e., that an internal observer cannot detect his absolute state of motion. (For earlier suggestions along these lines, see, for example,  and .)
The argument is the same for curved spacetimes; the internal observer would have no way to detect the “absolute” or fixed background. So the apparent background dependence provided by the (non-relativistic) condensed-matter system will not violate active diffeomorphism invariance, at least not for these internal inhabitants. These internal observers will then have no way to collect any metric information beyond what is coded into the intrinsic geometry (i.e., they only get metric information up to a gauge or diffeomorphism equivalence factor). Internal observers would be able to write down diffeomorphism invariant Lagrangians for relativistic matter fields in a curved geometry. However, the dynamics of this geometry is a different issue. It is a well-known issue that the expected relativistic dynamics, i.e., the Einstein equations, have to date not been reproduced in any known condensed-matter system.
Living Rev. Relativity 14, (2011), 3
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