Because these anchor points are on the same path, I cannot separate into different layers then lock one.
Even at full zoom, I cannot discern one point from the other and so only the "top" point gets selected.
I ended up having to mark the location of the point with intersecting guides, then dragging the upper point out of the way, then I could select and delete the point underneath, then drag the upper point back to the intersecting guides.
Although this solved my issue, it seems a bit klunky and I am hoping there is a better solution.
Obviously I can move and return the upper point successfully, whether by arrow key or by mapping the original point, but I am seeking a solution where I don't need to move the upper point at all. Why I cannot move the upper point is neither negotiable nor a topic for debate.