I agree that there are two systems in Schwarzschild coordinates. The system that has real solutions below the event horizon unambiguosly shows that light can travel from the event horizon to the central singularity and in the opposite direction from the singularity to the event horizon. If an event near the central singularity is connected to a future event just below the event horizon and the two events are within a common (and real) future light cone, then it seems suspect to be able to transform those events to another set of coordinates where the two events happen in the reverse order effectively reversing time.I wouldn't say this. I would say this that Schwarzschild coordinates really make up two different coordinates systems. One system is valid outside the black hole, but not on the event horizon or inside the black hole. The other system is valid inside the black hole, but not on the event horizon or outside the black hole. The metric unambiguously determines the nature of each coordinate in each coordinate system.
So, inside the event horizon, space doesn't get rotated into a time direction, it's just that human-chosen labels (coordinates) have names (chosen by humans, not by nature) that are very misleading. See