There is a no-go theorem forbidding flat and closed FLRW solutions in massive gravity on a flat reference metric, while open solutions are unstable. Recently it was shown that this no-go theorem can be overcome if at least some matter couples to a hybrid metric composed of both the dynamical and the fixed reference metric. We show that this is not compatible with the standard description of cosmological sources in terms of effective perfect fluids, and the predictions of the theory become sensitive either to the detailed field-theoretical modelling of the matter content or to the presence of additional dark degrees of freedom. This is a serious practical complication. Furthermore, we demonstrate that viable cosmological background evolution with a perfect fluid appears to require the presence of fields with highly contrived properties. This could be improved if the equivalence principle is broken by coupling only some of the fields to the composite metric, but viable self-accelerating solutions due only to the massive graviton are difficult to obtain.