We present a structure-preserving scheme based on a recently-proposed mixed formulation[1] for incompressible hyperelasticity formulated in principal stretches. Although there exits Hamiltonian introduced for quasi-incompressible elastodynamics using different formulations[2], the one in the fully incompressible regime has yet been identified in the literature. The adopted mixed formulation naturally provides a new Hamiltonian as an invariant for incompressible elastodynamics, which can be beneficial for the analysis of dynamic problems. Invoking the discrete gradient formulas, we are able to design fully-discrete schemes that preserve the Hamiltonian and momenta. The generalized Taylor-Hood element based on the spline technology provides a higher-order, robust, and inf-sup stable spatial discretization option for finite strain analysis. To enhance its performance in terms of volume conservation, the grad-div stabilization, a technique initially developed in computational fluid dynamics, is introduced here for elastodynamics. It is shown that the stabilization term does not impose additional restrictions for the algorithmic stress to be structure-preserving, leading to an energy-decaying and momentum-conserving fully discrete scheme. A set of numerical examples for Ogden-type materials is provided. In addition to the justification of discrete dissipation and conservation properties, the grad-div stabilization is found to enhance the discrete mass conservation effectively. Furthermore, in contrast to conventional algorithms based on Cardano’s formula and perturbation techniques, the spectral decomposition algorithm developed by Scherzinger and Dohrmann is robust and accurate to ensure the discrete conservation laws and is thus recommended for stretch-based material modeling.