Flutter - Webview with dropdown in the appbar - drop-down-menu

Question

I am trying to have a dropdown menu in a screen rendered by this webview library. The problem is that the rect drawn by this webview overlaps everything, as the picture below shows:
return new WebviewScaffold(
appBar: new AppBar(
backgroundColor: Colors.white,
title: new Theme(
child: new DropdownButton(
isExpanded: true,
items: <String>['Foo', 'Bar'].map((String value) {
return new DropdownMenuItem<String>(
value: value,
child: new Text(
value,
overflow: TextOverflow.ellipsis,
),
);
}).toList(),
onChanged: (_) {}),
data: new ThemeData.dark(),
),
url: new Uri.dataFromString("<h1>test</h1> <p>this is a test</p>",
mimeType: 'text/html',
parameters: {'charset': 'utf-8'}).toString(),
);
});
Then, I went further and I did this :
final flutterWebviewPlugin = new FlutterWebviewPlugin();
String url = "https://www.google.com";
double _height = 100.0;
GlobalKey _keyRed = GlobalKey();
Rect rect;
#override
void initState() {
super.initState();
flutterWebviewPlugin.launch(url, rect: rect);
}
void handlepress() {
setState(() {
RenderBox renderBoxRed = _keyRed.currentContext.findRenderObject();
_height = renderBoxRed.size.height;
print("height $_height");
});
}
#override
Widget build(BuildContext context) {
final mediaQuery = MediaQuery.of(context);
rect = new Rect.fromLTWH(
0.0, _height, mediaQuery.size.width, mediaQuery.size.height);
flutterWebviewPlugin.resize(rect);
return new AppBar(
backgroundColor: Colors.white,
title: new Theme(
child: new drop.DropdownButton(
key: _keyRed,
handler: handlepress,
isExpanded: true,
items: <String>['Foo', 'Bar'].map((String value) {
return new drop.DropdownMenuItem<String>(
value: value,
child: new Text(
value,
overflow: TextOverflow.ellipsis,
),
);
}).toList(),
onChanged: (_) => handlepress()),
data: new ThemeData.dark(),
),
leading: new IconButton(
icon: new Icon(
Icons.keyboard_arrow_left,
color: Colors.black,
size: 40.0,
),
onPressed: () => Navigator.of(context).pop(),
),
elevation: 0.0,
);
}
With this second code, drop.DropdownButton is my class that override DropDownButton in order to pass my Callback to onTap (click on the header of the dropdown). This code has effects on the rect rendered by the webview, but still do not work properly.
I also tried to hide the rect when the dropdown menu is extended, then show it again when it is collapsed. But there is not way I could find to detect these events.