  Methods for measuring the absolute center-of-mass energy using dileptons from
e$^{+}$e$^{-}$ collision events are further developed with an emphasis on
accelerator, detector, and physics limitations. We discuss two main estimators,
the lepton momentum-based center-of-mass energy estimator, $\sqrt{s}_{p}$,
discussed previously, and new estimators for the electron and positron
colliding beam energies, denoted $E^{\text{C}}_{-}$ and $E^{\text{C}}_{+}$. In
this work we focus on the underlying limitations from beam energy spread,
detector resolution, and the modeling of higher-order QED radiative corrections
associated with photon emissions originating from initial-state-radiation
(ISR), final-state-radiation (FSR), and their interference. We study the
consequent implications for the potential of these methods at center-of-mass
energies ranging from 90 GeV to 1 TeV relevant to a number of potential
accelerator realizations in the context of measurements of masses of the Z, W,
H, top quark, and new particles. The statistical importance of the Bhabha
channel for Higgs factories is noted. Additional extensive work on improving
the modeling of the luminosity spectrum including the use of copulas is also
reported.
