Due to high sensitivity, first-order perturbation analysis of structural-acoustic systems can be inaccurate for large perturbations. Using high-order derivatives to create a Taylor series approximation for the perturbed solution can result in slow convergence or even divergence. A rational polynomial or Padé approximation may overcome the poor convergence of the Taylor series by canceling out the poles causing the poor convergence. In this study, a finite element framework is used to describe the structural-acoustic system. External radiation and scattering problems are accommodated by truncating the infinite fluid region using exponential decay infinite elements. Changes to a nominal model are introduced through a structural perturbation to the nominal structural stiffness and/or mass matrices. An efficient method for calculating the solution derivatives with respect to the structural perturbation is presented. A Taylor series expansion is constructed using the derivative information and the convergence criteria for the series is examined. The local solution and derivatives are then used to construct a Padé approximation. The method is illustrated by considering what effect the addition of a rib stiffener to a plate in a rigid baffle has on the scattering of a plane wave. The approximation is shown to be quite accurate for large perturbations even when there are one or more nearby poles and the Taylor series fails to converge.