This paper presents RTD-RAX, a runtime-assurance extension of Reachability-based Trajectory Design that addresses two critical limitations of standard RTD implementations: conservatism from worst-case reachable-set overapproximations and inability to handle real-time disturbances. The framework combines a non-conservative RTD formulation for rapid goal-directed trajectory generation with mixed monotone reachability for fast disturbance-aware safety certification. When proposed trajectories fail safety verification under uncertainty, a repair procedure finds nearby safe trajectories that maintain progress toward the goal while guaranteeing safety under real-time disturbances. This approach enables provably safe, real-time trajectory planning for autonomous systems operating in uncertain environments.