Hourglass composite lattice sandwich beams (HCLSBs) possess excellent stiffness-to-weight efficiency, yet their dynamic vibration response requires further investigation. This paper establishes a modified Timoshenko beam model with bending-shear coupling for HCLSBs. The vibration response is systematically analyzed by the Galerkin truncation method (GTM) combined with the harmonic balance method (HBM), and the modal characteristics are validated through the finite element analysis and impact hammer experiments on fabricated physical specimens. On this basis, two distributed passive damping schemes based on nitinol steel wire rope (NiTi-ST) are designed and compared, including embedded and cell-based configurations. Vibration experiments demonstrate that both schemes effectively attenuate resonant responses. However, the NiTi-ST cell configuration exhibits superior damping efficiency on a mass-normalized basis. The proposed modeling framework and the distributed damping strategy provide an effective solution for vibration suppression in lightweight lattice structures, holding significant potential for applications in structural dynamics and control engineering, particularly in aerospace component design.