I can see that our plugin is not applying our format to the pagination, but the pagination is in the correct place in the HTML and defined correctly in the form.

That suggests to me that, in the PHP template for the page, the pagination is actually a call to an external function, rather than actually being run in the page template file. Are you able to show me that part of the PHP?

You would need to place code in that template to do the pagination without an external call like that. For example, if you had the free WP-PageNavi plugin, code like this would do it:

<div class="pagination">
	<?php
		if (function_exists('wp_pagenavi')) {
			echo "<br />";
			wp_pagenavi( array( 'query' => $query ) );
		}
	?>
</div>

Assuming that the wp_query results array is named $query, which is what most themes use (but not all).

You may need to go find the numeric_posts_nav() function and see if you can copy the code and place it directly in place in the template. But, it may be calling its own query, so that may get a little messy. You could write your own code, if you wish to avoid adding a plugin.

Possibly the $query part, if your theme uses a different name for that variable.

OR, did you make sure to end PHP before the HTML and start PHP after the HTML?

Do a recursive search through the theme files for wp_query( and see if you can find where the array is set.