Is there an add-on for a knife-like tool for curves? For instance, splitting a Bezier circle into cyclic halves cutting across its diameter is quite tedious when directly manipulating segments, control points and handles. Things get complicated when editing more complex curves with all the (un)hiding and moving of overlapping points to (re)connect and (re)align everything properly to keep the original shape intact.
No there is not. Unfortunately there are very few cad-like tools for bezier curves in blender. Curves are kind of a second-class citizen, and although they have improved considerably in recent years, they are still not regarded as primary modeling tool.

The closest I know you could currently get is probably a relatively recent addon "Extend / Trim Curve CAD Tools for Bezier Splines" which works relatively well for most situations:
http://blenderartists.org/forum/showthread.php?387520-Addon-Extend-Trim-Curve-CAD-Tools-for-Bezier-Splines