I've found a functional workaround. If I'm attempting to constrain the motion of a control point to a plane (and I effectively am), then all I need to do is select the connected four control points, and press ⇧ Shift + 1, ⇧ Shift + 3, or ⇧ Shift + 7 to constrain my view plane to the plane between them. Moving along this plane will be adequately close to the actual axis for my purposes.
I believe the reasons that this functionality is not yet implemented in Blender involve the lack of a mathematically sensible normal for a control point. Kind of like how NURBS is effectively built from rows, not vertices, and the rows are defined by control points; a control point in a vacuum is nonsense. This makes features like sliding a lot harder to define without pretending that something not-a-mesh is a mesh, which has its own problems.
But, arbitrary common points in space do have an arguable normal. So, this will do nicely.