This paper explores the modeling of stationary and oscillating discrete breathers in a biatomic crystal of the CsCl type with interatomic interactions described by the β-FPUT potential using molecular dynamics methods. Discrete breathers (DBs) are spatially localized oscillations of particles in defect-free nonlinear chains that can exist only in discrete media and nonlinear systems. Ordered CsCl-type structures belong to crystals with a body-centered cubic (BCC) lattice, the nonlinear dynamics of which have recently attracted considerable interest and attention from researchers. The history of the discovery and evolution of research into DBs and delocalized nonlinear vibrational modes (DNVMs) is presented. Growing interest in the study of mobile DBs capable of transferring energy across the crystal lattice is demonstrated. A computational cell for a bcc crystal with a CsCl structure is depicted, where the case of a large difference in the atomic masses of the components is analyzed. The DNVM and its frequency response are shown, where a gap appears in the phonon spectrum of the crystal. The frequency response of a discrete breather with a hard type of nonlinearity is presented, numerically found in the gap of the phonon spectrum by applying the localization function to the DNVM. This function has the form of a hyperbolic cosine, which allows the discrete breather frequency to deviate from the lower boundary of the phonon spectrum with increasing amplitude. Molecular dynamics simulations in the LAMMPS package resulted in discrete breathers centered on an atom and between two adjacent heavy atoms. Their motions are shown, where the atoms of the heavy sublattice move with much larger amplitudes than the light atoms. Finally, the time dependences of a stationary discrete breather centered on an atom and of discrete breathers oscillating near this configuration are shown. The possibility of the existence of discrete breathers only with an oscillation amplitude greater than 0.2 of the interatomic distance is noted.