Submitting a form added using ajax
I was able to solve this using the following code:

function bikebike_ajax_form()
{
    if (isset($_POST['url']))
    {
        $url = $_POST['url'];
        $menu_item = menu_get_item($url);
        $form = call_user_func_array($menu_item['page_callback'], $menu_item['page_arguments']);
        if (isset($form['#action']))
        {
            $form['#action'] = url($url);
        }
        $html = drupal_render($form);
        die($html);
    }
    die('');
}


As you can see, instead of rendering the page using file_get_contents, I simply grabbed the elements from the function defined in hook_menu. I needed to change the 'action' parameter of the resulting form structure.

If anyone has a better solution to this problem, I'll happily give you the answer.