How do I use radio buttons inside popup menus?

I'm trying to create a popup menu that contains selectable radio buttons in order to change a view type (e.g. gallery, cards, swipe, grid, list, etc.).

The issue I'm running into is that PopupMenu has its own callbacks for selecting values, and so does Radio and RadioListTile.

Ignore RadioListTile's onChanged

Here's my first attempt. This actually works, except that the buttons are perpetually grayed out. Giving the RadioListTiles a non-null noop function results in the buttons no longer grayed out (disabled), but then the popup menu no longer works.

new PopupMenuButton<String>(
    ...
    itemBuilder: (ctx) => <PopupMenuEntry<String>>[
          new PopupMenuItem(
              child: new RadioListTile(
                  title: new Text("Cards"),
                  value: 'cards',
                  groupValue: _view,
                  onChanged: null),
              value: 'cards'),
          new PopupMenuItem(
              child: new RadioListTile(
                  title: new Text("Swipe"),
                  value: 'swipe',
                  groupValue: _view,
                  onChanged: null),
              value: 'swipe'),
        ],
    onSelected: (String viewType) {
      _view = viewType;
    }));

Use RadioListTile, ignore PopupMenu

Second attempt is to ignore the PopupMenu entirely and just use RadioListTile onChanged. The buttons are not grayed-out/disabled, but are also not functional.

new PopupMenuButton<String>(
  ...
  itemBuilder: (ctx) => <PopupMenuEntry<Null>>[
        new PopupMenuItem(
            child: new RadioListTile(
                title: new Text("Cards"),
                value: 'cards',
                groupValue: _view,
                onChanged: (v) => setState(() => _view = v)),
            value: 'cards'),
        new PopupMenuItem(
            child: new RadioListTile(
                title: new Text("Swipe"),
                value: 'swipe',
                groupValue: _view,
                onChanged: (v) => setState(() => _view = v)),
            value: 'swipe'),
      ],
));

What's the correct approach? PopupMenu works well with extremely simple menus, but the element selection is giving me conflicts. Is there a way to get a "dumb" popup menu that displays a column of widgets (styled like a menu) at the button?