Multi-constrained routing is a key driver to support quality-of-service (QoS) for real-time multimedia applications in wireless mesh networks (WMNs). Due to the difficulty of applying strict admission control into a public WMN, it is inevitable to accommodate multiple application flows with different QoS requirements exceeding the capacity of a certain link shared by multiple flows. However, existing multi-constrained routing protocols under such an environment find the QoS degradation based on end-to-end path quality probing and trigger flooding-based route discovery from a scratch for resolving the QoS degradation, which incurs a longer recovery time and much routing overhead. In this paper, we propose a novel multi-constrained routing protocol for WMNs that finds problematic links that may affect QoS degradation to end-to-end paths and replaces them with a detour path using a local repair principle. We model congestion threshold estimation for finding problematic links and design algorithms for quickly finding detour paths and selecting an optimal path by minimizing the negative effect on existing flows nearby the detour path. Simulation results show that the proposed routing protocol achieves up to 19.6% more goodput of live video streaming applications with up to 33% reduced routing overhead compared with an existing work.