Copper busbar inserts are critical components of high-voltage connectors in new energy vehicles. The interfacial contact stress between the insert and the polymer directly affects the sealing reliability and service life of the connector. To address the interfacial stress concentration caused by the mismatch in thermal expansion coefficients between metal and polymer, this study employs COMSOL Multiphysics 6.2 simulations to investigate the regulation laws of arc-shaped and trapezoidal microstructures on the interfacial stress of copper–polyphenylene sulfide (PPS)/polypropylene (PP). The response surface methodology (RSM) is introduced to verify simulation reliability and optimize parameters. The simulation results indicate that both structures can effectively reduce interfacial stress, and the stress exhibits a significant nonlinear relationship with the structural parameters. Due to its high temperature resistance and polar thioether bond, PPS demonstrates better interfacial compatibility than PP. Under the same structural position, the maximum stress reduction exceeds 20% (from 0.689 MPa to 0.539 MPa). Moreover, the arc-shaped structure is more effective in alleviating stress concentration than the trapezoidal structure. At the same position, compared to the trapezoidal surface, the arc-shaped surface reduces the valley contact stress of PPS from 0.527 MPa to 0.5 MPa (a decrease of 5.12%) and that of PP from 0.679 MPa to 0.605 MPa (a decrease of 10.9%). The optimal parameters are as follows: an arc-shaped radius width of 1.0 mm, a depth of 0.8 mm; a trapezoidal bottom base of 2.0 mm, a height of 1.2 mm. This study provides a basis for the interface design of metal–polymer composite components and holds significant engineering value for the reliability optimization of high-voltage connectors.