Using a construction by Yamamoto of tropical contractions, we construct a non-archimedean SYZ fibration on the Berkovich analytification of a class of maximally degenerate hypersurfaces in projective space. We furthermore prove that under a discrete symmetry assumption, the potential for the non-archimedean Calabi-Yau metric is constant along the fibers of the retraction. The proof uses the work of Li on the Fermat degeneration of hypersurfaces, and an explicit description of toric plurisubharmonic metrics on the hybrid space associated to a complex toric variety.