Demand-responsive connectors have emerged as a promising solution to the first- and last-mile transit connectivity challenge, particularly for linking metro stations with surrounding communities. This paper investigates the optimal service and charging scheduling problem of electric vehicle demand-responsive connector (EV-DRC) with each vehicle serving multiple consecutive trips to minimize total cost (i.e., sum of the vehicle deployment and daily operational costs). This multi-trip electric vehicle routing problem (MTEVRP) is first formulated as an arc flow model, and then reformulated via Dantzig-Wolfe decomposition into a set-covering problem that can be solved by column generation. Subsequently, we propose an exact branch-and-price algorithm incorporating tailored acceleration strategies, including route classification, column selection, and interior point stabilization. Extensive computational experiments demonstrate the algorithm’s exceptional effectiveness - it achieves global optimality in 29 out of 30 test instances with up to 60 customer nodes, while reducing the runtime by a maximum of 71% compared to a standard branch-and-price. Sensitivity analyses reveal key insights regarding the impacts of vehicle capacity, battery specifications, and customer time windows on operational performance. These findings provide valuable guidance for EV-DRC operators to design economically viable DRC services that meet high standards of customer satisfaction.