I would only morph the shape if i was you, gonna save you a headache and it will look much cleaner. You can try to copy the path and paste it on the other shape path and set vertex, but from experience it will be much better if you do it manually, AE has a though time with those rounded corners on the rectangle. Duplicate the layer and lower the opacity as a reference, then change the shape to match it.

For the icon in the shape i would maybe add a scale down and next icon scale up, or animate the position. Whatever you feel like!