We propose a geometry-driven deterministic sampling method for Bingham distributions in arbitrary dimensions. With flexibly adjustable sampling sizes, the novel scheme can generate equally weighted samples that satisfy requirements of the unscented transform and approximate higher-order shape information of the Bingham distribution. By leveraging retraction techniques from Riemannian geometry, the sigma points are constrained to preserve the second-order moment. Meanwhile, samples in each principal direction are located in a way that minimizes a distance measure between the on-tangent-plane Dirac mixtures and the underlying on-manifold density. For that, the modified Cramér-von Mises distance based on the localized cumulative distribution (LCD) is employed. We further integrate the proposed approach into a quaternion-based orientation estimation framework. Compared to the existing unscented sampling approach drawing only fixed and limited numbers of sigma points, simulation results show that the proposed scheme enables better accuracy and robustness for nonlinear Bingham filtering.