Proper syntax or method for keeping url in modified isotope / category links

by

Use get_permalink (which returns the URL) instead of the_permalink (which prints the URL).

Try:

foreach ( $categories as $category ) {
    $link = get_term_link( $category );
    $html .= "<li><a href="https://wordpress.stackexchange.com/questions/159187/{$link}" data-filter="{$category->category_nicename}">{$category->cat_name}</a></li>";   
}