Drupal Bootstrap THEME_menu_tree on new menu
This isn't related to Bootstrap. In your preprocessor you're using theme_links, which is unrelated to menus. Your "menu" isn't being output as a menu at all. 

If page preprocessing is necessary, you can try:

function THEMENAME_preprocess_page(&amp;$vars) {
  $menu = menu_tree_all_data('menu-banner');
  $vars['banner_nav'] = menu_tree_output($menu);
}


Or you can render the menu block (these are automatically generated if you have the Blocks module enabled):

function THEMENAME_preprocess_page(&amp;$vars) {
  $block = block_load('menu', 'menu-banner');
  $vars['banner_nav'] = _block_get_renderable_array(_block_render_blocks(array($block)));
}


In page.tpl.php:

&lt;?php print render($banner_nav); ?&gt;


Either will work. 

If you don't need to preprocess the menu, you can simply position it via the Blocks admin UI. 

I hope that helps!