This paper investigates a class of stochastic Logistic harvesting models driven by tempered stable processes, with a one-sided power-law Lévy measure. We establish threshold conditions for population extinction and persistence, prove the distributional stability of the model, and derive explicit solutions for the optimal harvesting effort and the maximum sustainable yield. We systematically analyze the effects of white noise intensity and Lévy jump intensity on the optimal harvesting strategy. In particular, by focusing on the intrinsic structural parameters of the Lévy measure, namely the stability index and the tempering parameter, we elucidate their roles in shaping the optimal strategy and propose four targeted intervention strategies. Numerical simulations are presented to validate the theoretical findings.