After some time, I have figured out the answer to my own question. It was actually much easier than I thought. Hopefully, this will help others who may have wanted to do the same thing.
Go to "includes/index_filters/default_filter.php:
From what I can tell, this is where all the queries are made for the "Product Listing" module to display lists for both a specific category or manufacturer. By default, the current queries do not include "p.products_model". Simply add the column identifier in each query (4 total locations: category, category:all, manufacturer, manufacturer:all).
Example:
// We are asked to show only specific category
$listing_sql = "select " . $select_column_list . " p.products_id, p.products_model, p.products_type, p.master_categories_id, p.manufacturers_id, p.products_price, p.products_tax_class_id, pd.products_description, IF(s.status = 1, s.specials_new_products_price, NULL) as specials_new_products_price, IF(s.status = 1, s.specials_new_products_price, p.products_price) as final_price, p.products_sort_order, p.product_is_call, p.product_is_always_free_shipping, p.products_qty_box_status
from " . TABLE_PRODUCTS . " p left join " . TABLE_SPECIALS . " s on p.products_id = s.products_id, " .
TABLE_PRODUCTS_DESCRIPTION . " pd, " .
TABLE_MANUFACTURERS . " m, " .
TABLE_PRODUCTS_TO_CATEGORIES . " p2c
where p.products_status = 1
and p.manufacturers_id = m.manufacturers_id
and m.manufacturers_id = '" . (int)$_GET['filter_id'] . "'
and p.products_id = p2c.products_id
and pd.products_id = p2c.products_id
and pd.language_id = '" . (int)$_SESSION['languages_id'] . "'
and p2c.categories_id = '" . (int)$current_category_id . "'" .
$alpha_sort;
}
From there, you will be able to call the product model code "$listing->fields['products_model']" anywhere you would like, without the parameter in the admin back-end interrupting it from showing. In my case, it I included it directly under the product name in the "product_listing.php" file in modules.
Example:
$lc_text = '<h3 class="itemTitle"><a href="' . zen_href_link(zen_get_info_page($listing->fields['products_id']), 'cPath=' . (($_GET['manufacturers_id'] > 0 and $_GET['filter_id'] > 0) ? zen_get_generated_category_path_rev($_GET['filter_id']) : ($_GET['cPath'] > 0 ? zen_get_generated_category_path_rev($_GET['cPath']) : zen_get_generated_category_path_rev($listing->fields['master_categories_id']))) . '&products_id=' . $listing->fields['products_id']) . '">' . $listing->fields['products_name'] . '</a></h3><div id=\"listing_model\">Item Code: ' .$listing->fields['products_model'] . '</div>
Hope this helps someone. Why they made you originally have a whole column for the product model is beyond me, as both "Listing New" and "Listing All" both show it in the same column as the title and description.