Balancing the attainment of low-frequency bandgap with achieving higher load capacity is a significant concern in metamaterial design. By harnessing the post-buckling characteristics of bars, a novel tensegrity metamaterial is proposed, where the introduction of post-buckling leads to a softening of the structure"s stiffness, thereby enabling a low-frequency vibration isolation functionality with enhanced load-bearing capability. Utilizing the elliptic integral method to compute the post-buckling deformations of bars allows for the rapid determination of the stiffness of the tensegrity unit. Combined with the spring-mass diatomic chain model, bandgaps are calculated using Bloch"s theorem under periodic boundary conditions. To strike a balance between band gaps and load capacity, a data-driven dual-objective optimization method is employed, yielding the Pareto frontier of the post-buckling tensegrity metamaterial"s ultimate load and lower bandgap limit. Optimization results demonstrate that the bandgap frequency of the optimized structure can be as low as 3Hz, with a load capacity exceeding 100N. Compared to other low-frequency vibration isolation metamaterials, the ultimate load capacity can be increased by over 3.6 times at the same bandgap frequency.