How to add chunk of php, html, and sql code to variable an append to $content

You appear to be quite close to having this all sorted out. If you wrap your queries in a function, and separate the logic and presentation:

<?php 
function wpse_158425_get_terms(){
    global $wpdb, $rpt_options;

    $wpdb->show_errors();

    $rpt_days = $rpt_options['number_days'];

    $rpt_term_ids = $wpdb->get_col("
        SELECT term_id FROM $wpdb->term_taxonomy
        INNER JOIN $wpdb->term_relationships ON $wpdb->term_taxonomy.term_taxonomy_id = $wpdb->term_relationships.term_taxonomy_id
        INNER JOIN $wpdb->posts ON $wpdb->posts.ID = $wpdb->term_relationships.object_id
        WHERE DATE_SUB(CURDATE(), INTERVAL $rpt_days DAY) <= $wpdb->posts.post_date
        ");

        if(count($rpt_term_ids) > 0){

          $rpt_tags = get_tags(array(
            'orderby' => 'count',
            'order'   => 'DESC',
            'number'  => $rpt_options['number_tags'],
            'include' => $rpt_term_ids,
          ));

        } else {
            $rpt_tags = FALSE;
        }

        return $rpt_tags;

}

function wpse_158425_display_terms(){
    global $rpt_options;

    $tags = wpse_158425_get_terms();

    $recentTags="<div class="recent-popular-tags">";

    $recentTags .= '<h3>'.$rpt_options['heading_tags'].'</h3>';

    if($tags != FALSE){
        foreach($tags as $tag){
           $recentTags .= '<span class="rpt-link"><a href="' . get_tag_link ($rpt_tag->term_id) . '" rel="tag">' . $rpt_tag->name . '</a></span>';
        }
    } else {
            $recentTags .= 'There are no popular tags right now.';
    }

    $recentTags .= '</div><!--.recent-popular-tags-->';

    return $recentTags;

}

function rpt_display_content ($content){

    if(is_single()){    

        $rpt_display = wpse_158425_display_terms();    

        $content .= $rpt_display;
    }

    return $content;
}
add_filter('the_content', 'rpt_display_content');

?>

You can further simplify this by making wpse_158245_get_terms() accept variables and passing the options in that manner, and setting default variables.

I did not have a chance to run this, you might run into some object/array errors, but you should be good to go.