Dynamic multi-player games are powerful abstractions of important real-world problems involving multiple interacting agents in both cooperative and adversarial settings. This paper studies a three-player differential pursuit-evasion game in which a pursuer aims to capture a fleeing evader while a third player, the defender, cooperates with the latter by attempting to intercept or delay the pursuer to avoid capture. Our analysis considers time-varying dynamics and allows the presence of possibly moving obstacles in the domain. We apply a recent theoretical result to express the outcome of the game through the solution of a double-obstacle Hamilton-Jacobi-Isaacs variational inequality, and propose a novel approach to break down the problem into two simpler two-player games with dynamic targets and constraints, which can be solved at a much lower cost. Although conservative, this method guarantees correctness of the computed winning region and strategy for the evader-defender team when a feasible escape solution is found. We demonstrate both the full solution and the approximation method through a numerical example.