Low-frequency vibration control remains an important challenge in the design of engineered structures, as sufficient attenuation must be achieved while preserving structural integrity. In lattice metamaterials, this challenge becomes more complex due to a fundamental trade-off between vibration attenuation performance and mechanical properties. In particular, broadband vibration attenuation is often realized through discrete resonant elements, which may be accompanied by a reduction in mechanical stiffness. To alleviate this limitation, this study proposes a design optimization framework that enables broadband low-frequency vibration attenuation while maintaining lightweight and high stiffness. The strut geometry of the lattice unit cell is parameterized using Bézier curves, based on which a data-driven design framework is constructed by combining numerical dispersion analysis with Bayesian optimization. The relative phononic bandgap width computed from Bloch–Floquet wave analysis is adopted as the objective function for optimization. Numerical results show that the optimized Bézier-based lattice exhibits a significantly expanded low-frequency vibration attenuation region compared to conventional lattice designs. The normalized bandgap spans approximately from 0.23 to 1.03, corresponding to a bandwidth more than 40 times that of a uniform cylindrical lattice. In addition, the achieved bandgap width is approximately 6.9 and 4.1 times larger than those of cylinder–sphere and stepped-cylinder lattices, respectively. This enhancement is interpreted as being associated with changes in dynamic characteristics induced by continuous modulation of the strut geometry. Experimental transmission measurements conducted on additively manufactured lattice specimens reveal vibration attenuation behavior consistent with the numerically predicted band structure, despite finite-size effects. Quasi-static compression tests further confirm that the optimized lattice exhibits stiffness values approximately 17 and 3.4 times higher than those of representative discrete resonant lattice designs, respectively. These results suggest that continuous geometric modulation provides a mechanically reliable approach for achieving broadband low-frequency vibration attenuation while preserving load-bearing performance in lattice metamaterials.