This is actually quite possible, however, there are some specific things that you need to do to get it to work.
First, create a file on your wordpress installation that currently has that menu, for the demonstration, we’re calling it myoutsidermenu.php, in a theme called theme_name. Once you are done that, edit the php code so that it echoes your menu item into that file using the wordpress engine. I suppose you could include the CSS styling for the menu itself, but that isn’t entirely necessary. Remember, this file should only contain the menu, and no other wordpress elements, or they’ll be included on the receiving end which might cause you problems.
Once you are done that, on the receiving end, where you want your menu to show up, put the following code:
<?php
$doc = new DOMDocument();
$doc->loadHTMLFile('http://example.com/wp-content/themes/theme_name/myoutsidermenu.php');
echo $doc->saveHTML();
?>
Where the URL corresponds to the URL of the file that echoes the menu item. Now, remember to make sure to include CSS styling somewhere on your receiving end to make it display how you want it to display.
I’m not sure about any best-practices here, but it should get the job done.