How to conditionally add a wp_filter

Edit — fixed the if() statement.

function add_nav_class( $output ) {
    $menu_obj = get_term_by( 'slug', 'footer_menu', 'nav_menu' );

    // $menu_obj = get_term_by( 'name', 'Footer Menu', 'nav_menu' );
    // if the 'slug' version doesn't work, try the 'name' version
    // Just make sure you're using the correct name

    if( 'footer_menu' == $menu_obj->name ) ) {
        return $output; // bail if it's the 'footer_menu'
    }
    $output= preg_replace('/<a/', '<a class="lsbb"', $output, -1);
    return $output;
}

add_filter('wp_nav_menu', 'add_nav_class');

Edit 2 — replaced get_term() with get_term_by().

References