How do I make a skeletal limb respond to physical inertia?

Suppose I have a cat skeleton and have a animation running. My character will pick up the cat and shake it from side to side. How can I make the cat’s tail swing under the effect of inertia when shaking?

Fabrik IK and a manual swing value is the best.

AnimDynamics is second but not at all manual.

There’s a bunch of different solvers.

The best unattended way for physics is probably a physical animation profile and setting up the mesh to simulate all bodies below the attached piece of the tail…
This does have elevated costs.