Modified Newtonian Dynamics (MOND), postulating a breakdown of Newtonian mechanics at low accelerations, has considerable success at explaining galaxy kinematics. However, the quadrupole of the gravitational field of the Solar System (SS) provides a strong constraint on the way in which Newtonian gravity can be modified. In this paper we assess the extent to which modified gravity formulations of MOND are capable of accounting simultaneously for the Radial Acceleration Relation (RAR) -- encapsulating late-type galaxy dynamics -- the Cassini measurement of the SS quadrupole and the kinematics of wide binaries in the Solar neighbourhood. We achieve this by extending the method of Desmond (2023) to infer the location and sharpness of the MOND transition from the SPARC RAR under broad assumptions for the behaviour of the interpolating function and external field effect. We constrain the same quantities from the SS quadrupole, finding that it requires a significantly sharper transition between the deep-MOND and Newtonian regimes than is allowed by the RAR (an 8.7$\sigma$ tension under fiducial model assumptions). This may be relieved by allowing additional freedom in galaxies' mass-to-light ratios -- which also provides a better RAR fit -- and more significantly by removing galaxies with bulges. We show that the SS quadrupole constraint implies, to high precision, no deviation from Newtonian gravity in wide binaries in the Solar neighbourhood, and speculate on possible resolutions of this incompatibility between SS and galaxy data within the MOND paradigm.