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Abstract: A transducer capable of single-photon microwave-to-optical conversion would enable superconducting quantum computers to be networked together. We present our progress towards developing such a transducer by coupling a superconducting LC circuit and optical cavity using the mechanical mode of a silicon nitride membrane in a millikelvin environment. Our platform leverages the excellent mode control afforded by Fabry-Pérot cavities, allowing for field-leading transduction efficiency and continuous operation without disrupting the superconducting circuitry guiding the microwave mode. We present work in which we laser-cool the mechanical mode to its ground state, and discuss recent steps to develop the capability to demonstrate a quantum channel.