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 which is based on the so-called thin interface model and 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 and we introduce a new coupling framework which is now based on the so-called thick interface model. For scalar nonlinear hyperbolic equations in one space variable, we observe that the Cauchy problem is well-posed. Then, our main achievement in the present paper is the design of a new well-balanced finite volume scheme which is adapted to the thick interface model, together with a proof of its convergence toward the unique entropy solution (for a broad 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. Following a method proposed by Coquel and LeFloch, our proof relies on discrete entropy inequalities for the coupling problem and an estimate of the discrete entropy dissipation in the proposed scheme.