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We explore the kinematic scaling relations of 38 dwarf galaxies in the Fornax Cluster using observations from the SAMI integral field spectrograph. We focus on the Fundamental Plane (FP), defined by the physical properties of the objects (scale length, surface brightness, and velocity dispersion) and the Stellar Mass (Fundamental) Plane, where surface brightness is replaced by stellar mass, and investigate their dynamical-to-stellar-mass ratio. We confirm earlier results that the Fornax dEs are significantly offset above the FP defined by massive, hot stellar systems. For the Stellar Mass (Fundamental) Plane, which shows much lower scatter, we find that young and old dwarf galaxies lie at about the same distance from the plane, all with comparable scatter. We introduce the perpendicular deviation of dwarf galaxies from the Stellar Mass Plane defined by giant early-types as a robust estimate of their DM fraction, and find that the faintest dwarfs are systematically offset above the plane, implying that they have a higher dark matter fraction. This result is confirmed when estimating the dynamical mass of our dEs using a virial mass estimator, tracing the onset of dark matter domination in low mass stellar systems. We find that the position of our galaxies on the Stellar Mass FP agrees with the galaxies in the Local Group. This seems to imply that the processes determining the position of dwarf galaxies on the FP depend on the environment in the same way, whether the galaxy is situated in the Local Group or in the Fornax Cluster.