We provide a finite volume approximation in dimension $d \geq 1$ to a quasilinear parabolic equation with discontinuous hysteresis modelling a phase change, arising as a singluar limit of a pseudo-parabolic regularisation of a foward-backward diffusion equation. The convergence of the numerical solution to a suitable weak entropy solutions is shown under a parallelism assumption between the nonlinearities driving the evolution in each phase. The main challenge lies in the treatment of the discontinuous hysteresis operator in the proof of the compactness of the sequence of approximate solutions. This is achieved by regularising the hysteresis operator with a continuous one for which Hilpert inequalities are accessible and let us obtain crucial uniform translation estimates in $L^1$ in space. Numerical simulations, computed using a Julia-based framework for the finite volume discretisation of reaction-diffusion equations, are shown.