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The search for a physical model which explains the observed recent acceleration of the universe is a compelling task of modern fundamental cosmology. Recently Fernandes \textit{et al.} presented low redshift observational constraints on a scale invariant model by Maeder. Phenomenologically this can be interpreted as a bimetric theory with a time-dependent cosmological constant. It was shown that a matter density $Ω_m\sim0.3$ is a poor fit to the data, and the best-fit model would require a fluid with a much smaller density and a significantly positive equation of state parameter. This model is a particular case of an earlier and broader class of models by Canuto \textit{et al.}, which we study here. Specifically, we consider it in two distinct scenarios: as a genuine alternative to $Λ$CDM (i.e., without any cosmological constant) and as a parametric extension thereof (where both a cosmological constant and the new mechanism can coexist, and the relative contributions of both are determined by the data). We find that the first scenario can in principle fit the low-redshift data (but a good fit would require values of model parameters, such as the matter equation of state, in conflict with other data), while in the second one the deviation from $Λ$CDM is constrained to be small.