Do I need to use another custom post type for 1 post that differs from all of the others?

Here are some options I think you can try:

First Option:

If you always have that element in your archive page, you can add this element at the end of the list outside the loop. It should look something like this:

<?php
     $query = new WP_Query(array(
         'post_type' => array('team'),
         'orderby' => 'menu_order',
         'order' => 'ASC',
     ));

 if ($query->have_posts()) {
     while ($query->have_posts()) {
         $query->the_post(); ?>

         <li class="col-lg-3 d-flex text-center">
             <div class="team-member-wrapper text-center mx-auto">
                 <div class="img-border">
                 <?php
                     the_post_thumbnail('full', array('class' => 'team-member-img img-fluid'));
                 ?>
                 </div>
                 <?php
                     echo '<span class="team-member-position">' . get_post_meta(get_the_ID(), 'position', true) . '</span>';
                 ?>
             </div>
         </li>
                <?php
     }
     ?>
     <!-- This is where your last Send CV Element will go -->
     <?php
} else {

}
wp_reset_query();
wp_reset_postdata(); ?>

2nd Option

You can add a meta box in the post editor and create a send cv post and mark it as a send cv type. Then you can check for it in the front end.