Recent advances in tactile rendering span, among others, wearable cutaneous interfaces, tactile rendering algorithms, or nonlinear soft skin models. However, the adoption of these advances for multi-finger tactile rendering of dexterous grasping and manipulation is hampered by the computational cost incurred with nonlinear skin models when applied to the full hand. We have observed that classic constrained dynamics solvers, typically designed for contact mechanics, fail to perform efficiently on deformation constraints of nonlinear skin models. In this paper, we propose a novel constrained dynamics solver designed to perform well with highly nonlinear deformation constraints. In practice, we achieve more than 10x speed-up over previous approaches, and as a result we enable multi-finger tactile rendering of manipulation actions that capture the nonlinearity of skin.