Add filter to Custom Taxonomy Order NE plugin so that taxonomies list in customtaxorder_wp_get_object_terms_order_filter function can be modified.

customtaxorder_exclude_taxonomies.php

<?php
/*
 * customtaxorder_wp_get_object_terms_order_filter
 * wp_get_object_terms is used to sort in wp_get_object_terms and wp_get_post_terms functions.
 * get_terms is used in wp_list_categories and get_terms functions.
 * get_the_terms is used in the the_tags function.
 * tag_cloud_sort is used in the wp_tag_cloud and wp_generate_tag_cloud functions (but then the get_terms filter here does nothing).
 * Default sorting is by name (according to the codex).
 *
 */

function customtaxorder_wp_get_object_terms_order_filter( $terms ) {
	global $customtaxorder_settings;
	$options = $customtaxorder_settings;

	if ( empty($terms) || !is_array($terms) ) {
		return $terms; // only work with an array of terms
	}
	foreach ($terms as $term) {
		if ( is_object($term) && isset( $term->taxonomy ) ) {
			$taxonomy = $term->taxonomy;
		} else {
			return $terms; // not an array with objects
		}
		break; // just the first one :)
	}

	if ( !isset ( $options[$taxonomy] ) ) {
		$options[$taxonomy] = 0; // default if not set in options yet
	}
	if ( $options[$taxonomy] == 1 && !isset($_GET['orderby']) ) {
		if (current_filter() == 'get_terms' ) {
		  $customtaxorder_exclude_taxonomies = array('post_tag', 'product_tag');
			if ( in_array($taxonomy, apply_filters( 'customtaxorder_exclude_taxonomies', $customtaxorder_exclude_taxonomies )) ) {
				// no filtering so the test in wp_generate_tag_cloud() works out right for us
				// filtering will happen in the tag_cloud_sort filter sometime later
				// post_tag = default tags
				// product_tag = woocommerce product tags
				return $terms;
			}
		}
		usort($terms, 'customtax_cmp');
		return $terms;
	}
	return $terms;
}