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I think the line…

$post = $a_z_query->get_the_item_object();

…needs to in the While loop a little further down than line 54. I tried the same instructions but kept getting RETURNed with an empty/null value even querying inside the While loop. Eventually I edited the code below, starting at line 54, in a-z-listing.php in my child theme taking this…

<ul class="az-columns <?php echo esc_attr( $a_z_listing_column_class ); ?>">
	<?php
	while ( $a_z_query->have_items() ) :
		$a_z_query->the_item();
		?>
		<li>
			<a href="<?php $a_z_query->the_permalink(); ?>">
				<?php $a_z_query->the_title(); ?>
			</a>
		</li>
	<?php endwhile; ?>
</ul>

…and replacing it with this.

<ul class="az-columns <?php echo esc_attr( $a_z_listing_column_class ); ?>">
	<?php	
	while ( $a_z_query->have_items() ) :
		$a_z_query->the_item();
		$thumb = get_the_post_thumbnail_url($a_z_query->get_the_item_id(), 'thumbnail');
		?>
		<li>			
			<a href="<?php $a_z_query->the_permalink(); ?>"><img class="a-z-thumb" src='<?php echo $thumb; ?>' /><?php $a_z_query->the_title(); ?></a>
		</li>
	<?php endwhile; ?>
</ul>

Thanks a lot @mrchaotix! I got it to work.