How do I remove certain HTML elements with specific classes from the feed?

Use the filter the_content_feed. He is explizit for change content, before sending in feed.

A small example. Include a regex or replacement function to get your goal on content.

add_filter( "the_content_feed", "fb_example_change_content_feed" );
function fb_example_change_content_feed( $content ) {

   $content .= 'Total ' . str_word_count( $content ) . ' words';

   return $content;
}