We present comprehensive multicomponent dynamical models of M54 (NGC 6715), the nuclear star cluster of the Sagittarius (Sgr) dwarf galaxy, which is undergoing a tidal disruption in the Milky Way halo. Previous papers in this series used a large MUSE mosaic data set to identify multiple stellar populations in the system and study their kinematic differences. Here, we use Jeans-based dynamical models that fit the population properties (mean age and metallicity), spatial distributions, and kinematics simultaneously. They provide a solid physical explanation for our previous findings. Population-dynamical models deliver a comprehensive view of the whole system, and allow us to disentangle the different stellar populations. We explore their dynamical interplay and confirm our previous findings about the build-up of Sgr's nuclear cluster via contributions from globular cluster stars, Sgr inner field stars, and in situ star formation. We explore various parameterizations of the gravitational potential and show the importance of a radially varying mass-to-light ratio for the proper treatment of the mass profile. We find a total dynamical mass within M54's tidal radius (~75 pc) of 1.60 ± 0.07 × 10⁶ M _⊙ in excellent agreement with N-body simulations. Metal-poor globular cluster stars contribute about 65% of the total mass or 1.04 ± 0.05 × 10⁶ M _⊙. Metal-rich stars can be further divided into young and intermediate-age populations, which contribute 0.32 ± 0.02 × 10⁶ M _⊙ (20%) and 0.24 ± 0.02 × 10⁶ M _⊙ (15%), respectively. Our population-dynamical models successfully distinguish the different stellar populations in Sgr's nucleus because of their different spatial distributions, ages, metallicities, and kinematic features.