The scenario and gif sounds really similar to 2 Joints 2 limbs scenario however the method they use was 2D to 3D so the scenario ends up different.
The issue here is that the initial method to convert 3d into 2d is to project it into XY plane or XZ plane which well yeah only works in those two planes.
I believe the solution should be to resolve it to a plane that is relative to the crane system by using the parts right vector as a normal vector for the plane and the position vector of the first joint to form the plane of the arm. Closest point to plane projection should be the math term here, generally the way to convert 3d to 2d.
Otherwise, I believe I can whip up something similar using my CCDIK module so that may be an option if you want it.