You can recover the "perfect J" by phrasing the problem as "find the path that results in the monster having been waiting for you at the shore for the least time, or if he's too slow, that results in the maximum distance from the monster when you exit the lake". The best path is always the best path.

What really bothers me about the solution presented is that the (optimum) angle of escape is clearly exactly pi / 2, computed as the arcsine of 1. That's going to be exactly 1, but it's computed here as 4.603339 cos 1.351817, which is only approximate. There must be a solution that gives you the exact value; that's the one I want to see.