A class of particular interface problems, which is derived from Bazant-Storey-Kornyshev (BSK) theory to account for the electrostatic correlation in concentrated electrolytes, is studied in this paper. It involves a modified fourth-order Poisson-Fermi equation in solvents and a second-order Poisson equation in solutes with high-contrast coefficients, where nonhomogeneous interface conditions are introduced over the interface that divides solutes from solvents. A type of interface-fitted finite element method is developed and analyzed for this interface problem, and optimal error estimates are obtained for all variables in both $H^1$ and $L^2$ norms. Numerical experiments validate all attained theoretical results through two mathematical examples, as well as the electrostatic correlation phenomenon in concentrated electrolytes through a physical example, practically, where the electrostatic stress and interactional forces in the concentrated electrolyte are computed to reveal the charge reversal phenomenon that is governed by the BSK theory.