We determine the all-genus Hodge-Gromov-Witten theory of a smooth hypersurface in weighted projective space defined by a chain or loop polynomial. In particular, we obtain the first genus-zero computation of Gromov-Witten invariants for non-Gorenstein hypersurfaces, where the convexity property fails. We eventually extend it to any weighted projective hypersurface defined by an invertible polynomial.