Believe it or not, the answer is really simple

Try

  $menu->Post($x, $y);
[download]

instead of

  $menu->post($x, $y);
[download]

According to the Mastering Perl/Tk book (Steve Lidie & Nancy Walsh):

The post and Post Methods

The lower-level mechanism is the Menu post method, which posts a menu at a specific screen coordinate. The Post method works like post, but additioinally activates a specific menu item.

I tried... it worked.