Continuous-variable (CV) teleportation is a foundational protocol in quantum information science. A number of experiments have been designed to simulate ideal teleportation under realistic conditions. In our work, we detail an analytical approach for determining optimal input states for quantifying the performance of CV unidirectional and bidirectional teleportation. The metric that we consider for quantifying performance is the energy-constrained channel fidelity between ideal teleportation and its experimental implementation, and along with this, our focus is on determining optimal input states for distinguishing the ideal process from the experimental one. We prove that, under certain energy constraints, the optimal input state in unidirectional, as well as bidirectional, teleportation is a finite entangled superposition of twin-Fock states saturating the energy constraint. Moreover, we also prove that, under the same constraints, the optimal states are unique; that is, there is no other optimal finite entangled superposition of twin-Fock states.