In finite element procedures for dynamic problems in solid structures, explicit time integrators are commonly used for wave propagation calculations. However, their conditional stability requires determining time increments based on elastic wave propagation velocity. For nearly incompressible media such as rubber-like materials and biomaterials, explicit integrators face challenges with very small time increments due to high P-wave velocities representing volumetric deformations. Consequently, implicit time integrators with unconditional stability are preferred in such cases. The author proposed a novel time integrator with properties similar to explicit integrators in S-wave propagation for isochoric deformation. To address time increment limitations, the proposed scheme extends the Rattel method, designed for constrained Hamiltonian systems. The resulting method is a mixed time integration approach, incorporating an implicit solver for volumetric deformation propagation and an explicit solver for isochoric deformation. This study applies the method to large deformation problems in nearly incompressible hyperelasticity. For spatial discretization, the robust pressure-stabilized linear element is adopted.