Make child taxonomy categories use a set template

WordPress has a Template Hierarchy – specifically for taxonomies and terms. So what template name you want should follow along the lines of:

taxonomy-{$taxonomy}-{$term-slug}.php

In your case:

taxonomy-classifications-cheese.php

This will grab onto the cheese category and display this specific template. I’m not entirely sure if this will also catch child categories ( I would test the above first before trying the below ) but if it does not you can:

function cheese_template_redirect( $template ) {

    if ( is_tax( 'classifications' ) ) {
        $term   = get_term_by( 'slug', get_query_var( 'term' ), get_query_var( 'taxonomy' ) );
        $cheese = 13;   // This should be "Cheese" Category
        if ( term_is_ancestor_of( $cheese, $term->term_id, 'classifications' ) ) {
            $new_template = locate_template( array( 'taxonomy-classifications-cheese.php' ) );
            if ( '' != $new_template ) {
                return $new_template ;
            }
        }
    }

    return $template;
}
add_filter( 'template_include', 'cheese_template_redirect', 99 );

It looks like your issue is that you’re using the actual taxonomy rewrite slug, which we don’t want to use. I haven’t tested the above code but if it doesn’t work or if you run into issues, let me know.