We report the discovery of a new low-mass X-ray binary near the center of the unassociated Fermi GeV {gamma}-ray source 4FGL J0540.0-7552. The source shows the persistent presence of an optical accretion disk and exhibits extreme X-ray and optical variability. It also has an X-ray spectrum well-fit by a hard power law with {Gamma}=1.8 and a high ratio of X-ray to {gamma}-ray flux. Together, these properties are consistent with the classification of the binary as a transitional millisecond pulsar (tMSP) in the subluminous disk state. Uniquely among the candidate tMSPs, 4FGL J0540.0-7552 shows consistent optical, X-ray, and {gamma}-ray evidence for having undergone a state change, becoming substantially brighter in the optical and X-rays and fainter in GeV {gamma}-rays sometime in mid-2013. In its current subluminous disk state, and like one other candidate tMSP in the Galactic field, 4FGL J0540.0-7552 appears to always be in an X-ray "flare mode", indicating that this could be common phenomenology for tMSPs.