The dynamics of the Local Group (LG), especially concerning the contributions of the Milky Way (MW) and Andromeda (M31) galaxies, is sensitive to the presence of dark energy. This work compares the evolution of the LG by considering it as a two-body problem in a homogeneous and isotropic expanding spacetime, i.e. the Mc-Vitte spacetime (McV) versus the spherically symmetric metric for LG dynamics with the Cosmological Constant, i.e. the De Sitter-Schwarzschild spacetime (DsS). Using the Timing Argument (which links LG dynamics to LG mass) we find that the McV spacetime predicts a lower mass for the LG for the McV spacetime vs. the DsS spacetime. By the calibration of the IllustrisTNG simulations, the TA mass is shown to be biased high, and modified to be $~3 10^12$. With the Large Magellanic Cloud (LMC) the center of mass of the MW is shifted and reduces the mass the toal mass to $2.4 10^{12}$. The mass is compatible with other estimations in the literature, such as the Hubble flow and the Virial Theorem with the other dwarf galaxies in the LG and the full sum of the MW and M31 estimated mass.