The immersed boundary-finite element method (IBFE) is an approach to describing the dynamics of an elastic structure immersed in an incompressible viscous fluid. In this formulation, there are discontinuities in the pressure and viscous stress at fluid-structure interfaces. The standard immersed boundary approach, which connects the Lagrangian and Eulerian variables via integral transforms with regularized Dirac delta function kernels, smooths out these discontinuities, which generally leads to low order accuracy. This paper describes an approach to accurately resolve pressure discontinuities for these types of formulations, in which the solid may undergo large deformations. Our strategy is to decompose the physical pressure field into a sum of two pressure-like fields, one defined on the entire computational domain, which includes both the fluid and solid subregions, and one defined only on the solid subregion. Each of these fields is continuous on its domain of definition, which enables high accuracy via standard discretization methods without sacrificing sharp resolution of the pressure discontinuity. Numerical tests demonstrate that this method improves rates of convergence for displacements, velocities, stresses, and pressures, as compared to the conventional IBFE method. Further, it produces much smaller errors at reasonable numbers of degrees of freedom. The performance of this method is tested on several cases with analytic solutions, a nontrivial benchmark problem of incompressible solid mechanics, and an example involving a thick, actively contracting torus.