Flutter: Smoothly animate parent widget when child widget tree changes?

Question

I am attempting to build a button that, when tapped, will expand into a card that will contain a form for the user to fill in. My end goal is to have an animation similar to this for the button.

However, I am having trouble smoothly transitioning from one widget tree to another within a parent widget.

To visualise the change in the widget tree I am trying to achieve, consider that I want to go from the following state A...

Sscaffold:
    Column:
      - something_above
      - Card:
          Button
      - something_below

...to the goal state B...

Sscaffold:
    Column:
      - something_above
      - Card:
          Column:
            - TextField
            - DropdownMenu
            - Button
      - something_below

...all while smoothly animating the properties of the Card such as width and height.

My current attempt at such an animation consists of using an AnimatedContainer with a builder method for the child which returns either the original button or the form depending on some state parameter.

AnimatedContainer(
  child: _getChild(isOpen),
  duration: Duration(seconds: 1),
  color: Colors.blue,
)

However, this does not work and it appears that the animated container will only perform interpolation when one of it's own properties changes.

So it all comes down to the question; is there a widget that can facilitate this kind of transition, and if not where might I start to implement this kind of behavior?