Using a Corner Transfer Matrix approach, we compute the bipartite entanglement Rényi entropy in the off-critical perturbations of non-unitary conformal minimal models realised by lattice spin chains Hamiltonians related to the Forrester Baxter RSOS models in regime III. This allows to show on a set of explicit examples that the Rényi entropies for non-unitary theories rescale near criticality as the logarithm of the correlation length with a coefficient proportional to the effective central charge. This complements a similar result, recently established for the size rescaling at the critical point, showing the expected agreement of the two behaviours. We also compute the first subleading unusual correction to the scaling behaviour, showing that it is expressible in terms of expansions of various fractional powers of the correlation length, related to the differences Δ−Δmin between the conformal dimensions of fields in the theory and the minimal conformal dimension. Finally, a few observations on the limit leading to the off-critical logarithmic minimal models of Pearce and Seaton are put forward.