A surface acoustic wave (SAW) device, which is indispensable in a modern communication and sensing system, typically has a piezoelectric thin film deposited on an elastic substrate. As the operating frequency increases, the desired mode in such layered structures is increasingly accompanied by the spurious modes, which severely degrade the device performance. Although locally resonant (LR) metamaterials have shown potential for sub-wavelength wave manipulation, their coupling with layer-induced dispersion in piezoelectric layered media remains unexplored. Here, we establish an analytical model to reveal how this coupling governs the bandgap formation. By comparing the high-velocity (AlN-Si) and low-velocity (LiNbO₃-Si) systems, we uncover two distinct mechanisms: the high-velocity system enables a complete bandgap; in the low-velocity system, higher-order modes traverse the resonant frequencies without the classic avoided-crossing behavior, preventing the complete bandgap formation. These findings establish quantitative design guidelines for the selective spurious mode suppression in next-generation SAW devices.