We continue our analysis of the coupling between nonlinear hyperbolic problems across possibly resonant interfaces. In the first two parts of this series, we introduced a new framework for coupling problems, based on the so-called thin interface model, which uses an augmented formulation and an additional unknown for the interface location; this framework has the advantage of avoiding any explicit modeling of the interface structure. In the present paper, we pursue our investigation of the augmented formulation but introduce a new framework for coupling problems, now based on the so-called thick interface model. For scalar nonlinear hyperbolic equations in one space variable, we observe first that the Cauchy problem is well-posed. Our main achievements in the present paper are, on one hand, the design of a new well-balanced finite volume scheme which is adapted to the thick interface model and, on the other hand, a proof of the convergence of this scheme toward the unique entropy solution to the Cauchy problem for a large class of nonlinear hyperbolic equations. Due to the presence of a possibly resonant interface, the standard technique based on a total variation estimate does not apply, and DiPerna's uniqueness theorem must be used instead. Our proof relies on discrete entropy inequalities for the coupling problem and an estimate of the entropy dissipation of the proposed discrete scheme.