  The polaron concept captures physical situations involving an itinerant
quantum particle (excitation) that interacts strongly with bosonic degrees of
freedom and becomes heavily boson-dressed. While the Gerlach-L\"{o}wen theorem
rules out the occurrence of nonanalyticities of ground-state-related quantities
for a broad class of polaron models, examples were found in recent years of
sharp transitions pertaining to strongly momentum-dependent interactions of an
excitation with dispersionless (zero-dimensional) phonons. On the example of a
lattice model with Peierls-type excitation-phonon interaction, such
level-crossing-type small-polaron transitions are analyzed here through the
prism of the entanglement spectrum of the excitation-phonon system. By
evaluating this spectrum in a numerically-exact fashion it is demonstrated that
the behavior of the entanglement entropy in the vicinity of the critical
excitation-phonon coupling strength chiefly originates from one specific
entanglement-spectrum eigenvalue, namely the smallest one. While the discrete
translational symmetry of the system implies that those eigenvalues can be
labeled by the bare-excitation quasimomentum quantum numbers, here it is shown
numerically that they are predominantly associated to the quasimomenta $0$ and
$\pi$, including cases where a transition between the two takes place deeply in
the strong-coupling regime.
