Text with special class as excerpt (or exclude special class)

The easiest course of action would be to write manual excerpts instead of having the system generate them.

Second, would be to wrap your “content that isn’t an excerpt” in a shortcode. If you look at the wp_trim_excerpt() function, which generates excerpts from post content, one of the first thing it does is strip shortcodes. For example, if you had a shortcode like this:

function foocode($atts,$content){
  return $content;
}
add_shortcode('foo','foocode');

Anything in the post body such as [foo]Hello[/foo] would not appear in the generated excerpts. Note that the “exceprt” is not necessarily the chunk of text that displays on the index page. If the the theme uses the_content() the post content will break on the <!-- more --> tag, or nothing at all, and your content will still appear.