Time-Sensitive Networking (TSN) is a promising solution for the next generation in-vehicle networks. To enable innovations in adaptability, there is a strong trend to integrate TSN with Software-Defined Networking (SDN). Software-defined TSN requires fast routing algorithms that guarantee the Worst-Case End-to-End Delay (WCED) of the Audio Video Bridging (AVB) flows. However, current AVB flow routing algorithms execute WCED analysis for an exponential amount of route candidates, which is not feasible at run-time. Current WCED analysis of the AVB flows depends on the specific flow setup of other traffic classes. When high-priority flows change, low-priority flows must be adjusted as well to maintain deadline guarantees. In this paper, we propose a per-class flow management scheme to enable run-time routing of the AVB flows. Our composable analysis computes a formal WCED bound of the AVB flows. It does not rely on the flow setup of other traffic classes. So, the WCED and routes can be computed for each traffic class independently and in parallel. Based on the analysis, we develop a fast heuristic algorithm to incrementally route the AVB flows with guaranteed deadlines. Our experiments demonstrate that, compared to the existing solutions, the composable analysis is 2.6x faster. The cost is that the WCED bound can be up to 31% higher. When given a similar run-time as existing solutions, the proposed routing algorithm can set up 1.6x more flows. The average time to set up one flow is within 0.14s.