  We investigate spectral properties of polaronic excitations within the
framework of an analog quantum simulator based on inductively coupled
superconducting transmon qubits and microwave resonators. This system emulates
a lattice model that describes a nonlocal coupling of an itinerant
spinless-fermion excitation to dispersionless (Einstein-type) phonons through
the Peierls and breathing-mode interaction mechanisms. The model is
characterized by a sharp, level-crossing transition at a critical value of the
effective excitation-phonon coupling strength; above the transition point, the
ground state of this model corresponds to a heavily dressed (small-polaron)
excitation. Using the kernel-polynomial method, we evaluate the
momentum-frequency resolved spectral function of this system for a broad range
of parameters. In particular, we underscore the ramifications of the fact that
the zero-quasimomentum Bloch state of a bare excitation represents the exact
eigenstate of the Hamiltonian of this system for an arbitrary excitation-phonon
coupling strength. We also show that -- based on the numerically evaluated
spectral function and its well-known relation with the survival probability of
the initial, bare-excitation Bloch state (the Loschmidt echo) -- one can make
predictions about the system dynamics following an excitation-phonon
interaction quench. To make contact with anticipated experimental realizations,
we utilize a previously proposed method for extracting dynamical-response
functions in systems with local (single-qubit) addressability using the
multiqubit (many-body) version of the Ramsey interference protocol.
