The one-dimensional system of elasticity with a non-monotone or convex-concave stress-strain relation provides a model to describe the longitudinal dynamics of solid-solid phase transitions in a bar. If dissipative effects are neglected it takes the form of a system of first-order nonlinear conservation laws and dynamical phase boundaries appear as shock wave solutions. In the physically most relevant cases these shocks are of the non-classical undercompressive type and therefore entropy solutions of the associated Cauchy problem are not uniquely determined. Important dissipative effects that lead to unique regular solutions are viscosity and capillarity where the latter effect is usually modelled by at least third-order spatial derivatives.\\ Differently from these models we consider a novel type of non-local regularization that models both effects but avoids high-order derivatives. We suggest a particular scaling for the dissipative terms and conjecture that with this scaling the regular solutions single out unique physically relevant weak solutions of the first-order conservation law in the limit of vanishing dissipation parameter. We verify the conjecture first by proving that the non-local system admits special solutions of traveling-wave type that correspond to dynamical phase boundaries. Moreover it is proven that regular solutions of a general Cauchy problem converge to weak solutions of the system of first-order conservation laws. The proof is achieved by the method of compensated compactness.