For mixed-criticality (MC) systems, recent studies show that it can be important to provide continuous (albeit degraded) services for low-critical (LC) tasks even in the high running mode. In this paper, focusing on dual-criticality systems, we study a mode-switch fixed-priority (MS-FP) scheduler for a set of dual-rate mixed-criticality (DR-MC) tasks, where each LC task can have a pair of small and large periods to represent its service requirements in the low (LO) and high (HI) running modes, respectively. Moreover, DR-MC tasks may adjust their priorities at the mode-switch point for better system schedulability. By extending the response time analysis (RTA) technique for MC systems, we first derive the schedulability conditions for a set of DR-MC tasks under the MS-FP scheduler with mode transition being considered. Then, we investigate how to select periods and priorities of DR-MC tasks to optimize their control performance and formulate it as a Non-Linear Optimization problem. We propose an efficient heuristic for a simplified optimization problem based on Branch a Bound Search Tree (BBST) technique. The effectiveness of the proposed heuristic and the MS-FP scheduler with DR-MC task model is illustrated through one case study with four tasks and compared against the Ipopt solutions.