So I went ahead and did the multiple loop thing, as I had done before. I'm always interested in seemingly cleaner/simpler solutions, but sometimes you just have to keep moving. This is the code I'm using:
$args = array(
'post_type' => 'projects',
'project_category' => 'websites',
'orderby' => 'menu_order',
'showposts' => 1
$posts = get_posts( $args );
foreach ($posts as $post) : setup_postdata($post);
// Get the post stuff here
<?php endforeach; wp_reset_postdata(); ?>
I can repeat this as needed, changing the taxonomy (in this case the taxonomy is project_category) term each time.
Please feel free to comment an tell me if this isn't a good way to do it. I'm far from an expert at the PHP side of WordPress.