We investigate the nonlinear Schrödinger equation on a three-edge star graph, where each edge contains a localized inhomogeneity in the form of a Dirac delta linear potential. Such systems are of significant interest in studying wave propagation in networked structures, with applications in, e.g., Josephson junctions. By reducing the system to a set of finite-dimensional coupled ordinary differential equations, we derive explicit conditions for the occurrence of a symmetry-breaking bifurcation in a symmetric family of solutions. This bifurcation is shown to be of the transcritical type, and we provide a precise estimate of the bifurcation point as the propagation constant, which is directly related to the solution norm, is varied. In addition to the symmetric states, we explore non-positive definite states that bifurcate from the linear solutions of the system. These states exhibit distinct characteristics and are crucial in understanding solutions of the nonlinear system. Furthermore, we analyze the typical dynamics of unstable solutions, showing their behavior and evolution over time. Our results contribute to a deeper understanding of symmetry-breaking phenomena in nonlinear systems on metric graphs and provide insights into the stability and dynamics of such solutions. © 2025 Elsevier B.V.