You can only retrieve your UserControl by name from the template, so you'll need to supplement your UserControl with the following method:

public ToggleButton FindToggleButton()
{
    //since the ToggleButton inside the control has a name ("PART_ToggleButton"), we can simply access it via a field of the same name
    return PART_ToggleButton;
}

Just put this method in your control's code-behind.

Then you need to modify the method retrieving the ToggleButton from the ListView:

...
//first we retrieve the GroupItem control associated with current second-level group
var container = listView.ItemContainerGenerator.ContainerFromItem(leagueGroup) as GroupItem;
//then we find the named FavouriteControl inside the template
var favouriteControl = container.Template.FindName("PART_FavouriteControl", container) as FavouriteControl;
var toggleButton = favouriteControl.FindToggleButton();

Note that to avoid confusion I used the name "PART_FavouriteControl", so you need to update your GroupItem template accordingly, so that it has:

<LSControls:FavouriteControl x:Name="PART_FavouriteControl" (...)>