The full Born-Oppenheimer (BO) potential energy surface that describes the interaction between a vibrating and rotating LiH molecule and an He atom, obtained from ab initio VB calculations described before (Chem. Phys. 1997, 215, 287), is employed here to evaluate the corresponding partial integral cross sections over a range of energies of astrophysical relevance. The process examined involves rotovibrational heating of the initially "cold" target molecule in a single-collision situation, and it provides specific indications on the process efficiency for modeling global kinetics of primordial clouds. The details of the dynamical coupling produced and some specific features of the potential energy surface are analyzed in relation to the behavior of the inelastic cross sections. The average energy transfer values and the relaxation rates are obtained from quantum dynamics and found to be unexpectedly large for such a weakly interacting system.