In this paper we propose a procedure for estimating the region in which a controller robustly stabilizes a system which is subject to affine parametric uncertainty by applying transverse contraction-based stability tools. The method consists of an optimization problem in which transverse contraction conditions are verified via sum-of-squares programs. The optimization approach can be used either to maximize the bounds on the allowable parameter uncertainty or to maximize the size of the region of contraction (ROC) given a fixed level of uncertainty. In a case study we apply the procedure to an Airborne Wind Energy system where the flight path of a power generating kite is controlled by a linear quadratic regulator based on a model which is prone to large parametric uncertainties. We consider periodic trajectories of the stabilized kite system and transform the dynamics into transversal coordinates for simplification of the controller design and reduction of the computational cost. The numerical results of the proposed optimization show that uncertainty in the steering gain parameter decreases the size of the ROC while uncertainty in wind speed or line length within the considered range of operating conditions does not affect the size of the robust ROC.