Hey, sorry about the delay..
The CCDIK and FABRIK solvers in FIK can't solve this right, as both algorithms can only solve for the position of the target, not the rotation.
It might be possible to solve it with Unity physics instead though. It is possible to set up physics to make it work sort of like an inverse dynamics solver. Here's a package I made - Inverse Dynamics.
Basically just set up the rig with Rigidbodies, Joints and optionally colliders as if it was a ragdoll. Then attach the root to a kinematic rigidbody and also the last part to a kinematic rigidbody - that will be your end effector. Then add InverseDynamics.cs to the root kinematic rigidbody. You can use joint limits and colliders for internal and/or external collisions if you need to, I have managed to use this to solve all kinds of complex rigs with prismatic joints and what not using this tool.