Optomagnonic cavities have recently been emerging as promising candidates for implementing coherent photon-magnon interactions, for applications in quantum memories and devices, and next generation quantum networks. A key challenge in the design of such cavities is the attainment of high magnon-mediated optical-to-optical conversion efficiencies, which could, e.g., be exploited for efficient optical interfacing of superconducting qubits, as well as the practicality of the final designs, which ideally should be planar and amenable to on-chip integration. Here, on the basis of a novel time-Floquet scattering-matrix approach, we report on the design and optimization of a planar, multilayer optomagnonic cavity, incorporating a cerium-substituted yttrium iron garnet thin film, magnetized in-plane, and operating in the triple-resonant inelastic light scattering regime. This architecture allows for magnon-mediated optical-to-optical conversion efficiencies of about 5% under realistic conditions, which is orders of magnitude higher than that attained in alternative optomagnonic designs. Our results suggest a viable way forward for realizing practical information inter-conversion, with high efficiencies, between microwaves, strongly coupled to magnons, and optical photons, as well as a platform for fundamental studies of classical and quantum dynamics in magnetic solids and for the implementation of futuristic quantum devices.