I will reply in the meantime with most likely the least correct solution, until someone with more curve experience can offer better. As far as I understand, unless it is a bezier it doesn't have a proper editable tangent. What about this?
1. Select the target edit points: (P-1) , P , (P+1)
2. detachCurve (now you have 4 curve components)
3. Select the two curves: (P-1, P) and (P, P+1)
4. rebuildCurve: uniform, Keep Ends, Spans 1, Degree 1
5. Select all curves
6. attachCurve: connect, keep multiple knots
Ok now someone else can point out that I have probably overcomplicated this and there is a simpler way.
Justin