Think I worked it out!!
I did a print_r on each $menu_item and saw there’s an array key called menu_item_parent in there.
So I changed this:
foreach ( (array) $menu_items as $key => $menu_item ) {
$title = $menu_item->title;
$url = $menu_item->url;
$menu_output .= '<option value="' . $url . '">' . $prefix . $title . '</option>';
}
to this:
foreach ( (array) $menu_items as $key => $menu_item ) {
if ( $menu_item->menu_item_parent == 0 ) :
$title = $menu_item->title;
$url = $menu_item->url;
$menu_output .= '<option value="' . $url . '">' . $prefix . $title . '</option>';
endif;
}
Now it will only pull menu items without a menu item parent.