Specific template for subcategory of custom taxomy

You could use the template_include hook to load a specific template for your taxonomy. You would need to change out any instances of my-taxonomy with your specific custom taxonomy and switch out the brand-subcategories.php with whatever your template name is.

/**
 * Force Taxonomy Subcategories to new template
 *
 * @param String $template - Expected template path
 *
 * @return String $template
 */
function wpse303717_subcategory_template( $template ) {

    // We're not on a taxonomy page
    if( ! ( is_tax() && is_tax( 'my-taxonomy' ) ) ) {
        return $template;
    }

    // Grab the queried object, _should_ be a term but make sure.
    $queried_object = get_queried_object();

    // We either don't have the right object OR we're on a top level category becuase $term->parent === 0
    if( isset( $queried_object->parent ) && empty( $queried_object->parent ) ) {
        return $template;
    }

    // Our template could be located anywhere, we could store it in a subdirectory but 
    // we would need to specify a relative path from the theme root to it.
    // Example: 'templates/brand-subcategories.php'
    if( false !== ( $new_template = locate_template( 'brand-subcategories.php' ) ) ) {
        $template = $new_template;
    }

    return $template;

}
add_filter( 'template_include', 'wpse303717_subcategory_template' );

If you want to load a specific template per subcategory you would need to test against $queried_object->term_id vs what you expect. Though do note that your client could delete these terms which would throw off your code.