[Need Help] How to get stretchy bones behaviour with Control Rig

I am trying to add a stretch behaviour to the bones with the control rig, while maintaining the volume of the bone. Something like the “Stretch To” Constraint in Blender.

I created a video that demonstrates what I am looking for, the steps I am following to get it done, and the issues that I am facing when exporting the animation.

The goal is to create this behaviour and be able to export the correct animation for use in other software, like Blender or Godot.

Here is the link to the video: https://www.youtube.com/watch?v=QZrUJloE_Fw