Idiomatic way to change aspects of the active scaffold from within Flutter TabView child

Question

Essentially, I'm trying to devise a way for a child of a TabView to customise aspects of the active Scaffold instance, such as the floating action button, application bar and so on.

Using the code snippet below, the idea is for_ATabState to set the floatingActionButton of the Scaffold instance to a custom widget that it controls.

class MasterWidget extends StatefulWidget {
  @override
  _MasterWidgetState createState() => _MasterWidgetState();

  Widget? setFloatingActionButton() {
    // how to access state and invoke `setState`?
  }
}

class _MasterWidgetState extends State<MasterWidget> {
  @override
  Widget build(BuildContext context) {
    final tabs = [ATab()];

    return Scaffold(
      appBar: AppBar(),
      floatingActionButton: activeTab.buildFloatingActionButton(),
      body: DefaultTabController(
        length: tabs.length,
        child: TabBarView(
          children: tabs,
          controller: tabController,
        ),
      ),
    );
  }
}

class ATab extends StatefulWidget {
  @override
  _ATabState createState() => _ATabState();
}

class _ATabState extends State<ATab> {
  @override
  Widget build(BuildContext context) {
    // Doesn't work:
    // context.findAncestorWidgetOfExactType<Scaffold>()?.floatingActionButton = AFloatingActionButton();
    // context.findAncestorWidgetOfExactType<MasterWidget>()?.setFloatingActionButton(AFloatingActionButton());

    return SomeWidget();
  }
}

Here's what I tried:

1. Try to context.findAncestorWidgetOfExactType<Scaffold>() in _ATabState and somehow set Scaffold's floatingActionButton attribute; unfortunately there does not seem to be a setter available.

2. Try to context.findAncestorWidgetOfExactType<MasterWidget>() in _ATabState but then I'm not able to access the state where the rendering takes place.

What's the approach applicable here?