We study the coupling of two conservation laws with different fluxes at the interface $x=0$. The coupling condition yields (whenever possible) continuity of the solution at the interface. Thus the coupled model is not conservative in general. This gives rise to interesting questions such as nonuniqueness of self-similar solutions which we have chosen to analyze via a viscous regularization. Introducing a color function, we rewrite the problem in a conservative form involving a source term which is a Dirac measure. In turn, this leads to a nonconservative system which may be resonant. In this work, we analyze the regularization of this system by a viscous term following Dafermoss approach. We prove the existence of a viscous solution to the Cauchy problem with Riemann data and study the convergence to the solution of the coupled Riemann problem when the viscosity parameter goes to zero.