We present a theoretical study of nonreciprocal charge transport in two-dimensional noncentrosymmetric superconductors, taking the transition-metal dichalcogenide MoS₂ as a representative example. In the normal state, the magnetochiral anisotropy vanishes within the minimal band model of MoS₂, appearing only at subleading order in the symmetry-breaking perturbations set by trigonal warping, Ising spin-orbit coupling, and the Zeeman field. Superconductivity changes this picture qualitatively: in the vicinity of the transition, the magnetochiral anisotropy is strongly enhanced by pairing fluctuations. We evaluate the nonreciprocal current density arising from order-parameter fluctuations and quantum-interference processes – the Aslamazov-Larkin and Maki-Thompson channels – and show that both are governed by cubic Lifshitz invariants of the Ginzburg-Landau free energy, generically allowed once inversion and time-reversal symmetries are broken. These invariants are derived microscopically from the band model, including the effects of disorder: in the diffusive limit the warping-induced invariant is suppressed, yet the resulting response remains sizable. Strain is shown to enable additional vector components of the nonlinear current, activating the Maki-Thompson channel. Finally, invoking Onsager reciprocity, we identify kinetic Lifshitz invariants, nonreciprocal corrections to the order-parameter relaxation rate, locked to the Langevin noise by the fluctuation-dissipation theorem, and demonstrate that their contribution to the magnetochiral anisotropy is parametrically subleading near the transition.