  Using a Corner Transfer Matrix approach, we compute the bipartite
entanglement R\'enyi 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 R\'enyi 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 $\Delta-\Delta_{\min}$ 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.
