Orientation in 2D

[_Aka_]

That seems to affect control points but not orientation.

Yes, Restrict to 2D affects only CPs.This option doesn't modify the inner workings of the spline, but instead just adds restrictions to the editor to make it easier to work on a single plane.

Likewise, changing my CurvySpline orientation from Dynamic to Static, or None, also does not get what I want, although leaves sprite in the correct plane.

You can set the orientation of each control point the way you want by setting the spline's orientation to Static and then ticking the Bake Orientation option for each control point. You will then be able to define the orientation at a control point by modifying the control point's rotation.

If my answer is not helpful, please send me a scene, or at least screenshots, so I can see what you are trying to do.