JavaFX: set fullscreen exclusive mode (FSEM) in non-mouse/keyboard-handler

Question

my application consists of a TabPane named tabPane with, lets say, two Tabs: firstTab and secondTab. Whenever firstTab is selected, I want the application to be in FSEM (full-screen exclusive mode, see documentation); otherwise it should be in normal windowed mode.

My usual way of being notified about which Tab is active is either

BooleanBinding isFirstTabActive = tabPane.getSelectionModel().
    selectedItemProperty().isEqualTo(firstTab);

or

tabPane.getSelectionModel.selectedItemProperty().addListener(...);

Since stage.fullScreenProperty() is read-only and therefore cannot be binded (as stated in docs), I can not use my first method and do, lets say,

stage.fullScreenProperty().bind(isFirstTabActive) //not working.

Since I am not planning on sigining the application for now and, as the docs state:

Applications can only enter FSEM in response to user input. More specifically, entering is allowed from mouse (Node.mousePressed/mouseReleased/mouseClicked) or keyboard (Node.keyPressed/keyReleased/keyTyped) event handlers.

I can not use the second method either, by doing

tabPane.getSelectionModel.selectedItemProperty().addListener(new ChangeListener<Tab>() {
  @Override
  public void changed(ObservableValue<? extends Tab> observable, Tab oldValue, Tab newValue) {
    stage.setFullScreen(newValue == firstTab);
  }
);

because this way the application does not enter FSEM but only emulated fullscreen mode.

I do know that I can enter FSEM by creating EventHandlers which listen to tabPane events. But this way I had to add two EventHandlers for both MouseEvent.MOUSE_CLICKED and KeyEvent.KEY_TYPED, and the EventHandlers would have to explicitly find out whats going on ignoring their Event, like:

tabPane.addEventHandler(MouseEvent.MOUSE_CLICKED, new EventHandler<MouseEvent>() {
  @Override
  public void handle(MouseEvent event) {
    stage.setFullScreen(tabPane.getSelectionModel().getSelectedItem() == firstTab);
  }
});

So is this really the way to go here? Or can I do some workaround with the Bindings? Any suggestions here? Thanks.