  We use the Momentum Average approximation together with perturbative
approaches, in the appropriate limits, to study the single polaron physics on a
perovskite lattice inspired by BaBiO3. We investigate electron-phonon coupling
of the Peierls type whereby the motion of ions modulates the values of the
hopping integrals between sites, and show that it cannot be mapped onto the
simpler one-band Holstein model in the whole parameter space. This is because
the dispersion of the Peierls polaron has sharp transitions where the
ground-state momentum jumps between high-symmetry points in the Brillouin zone,
whereas the Holstein polaron always has the same ground-state momentum. These
results imply that careful consideration is required to choose the appropriate
model for carrier-lattice coupling in such complex lattices.
