Facing issue in offset path in after effect

So you're using offset path to animate a shape to move across a path, but I'm guessing you don't want the path to start out with a square end.   I think the simplest solution is to lengthen the path and start the animation with the shape rounded (e.g. 5%).  if you want the length of the shape to increase, I think you'll need to keyframe that separately.

What are you trying to do that you can't do? I just see a mouse pointer wiggling around over an animated shape layer.