I want to extend the current themes’ single.php to display the meta fields of my CPT

Well, you could hook into the the_content filter, asking if it was a single custom post type and then replace the output with something you like.

Something like this:

function content_my_cpt_filter( $content ) {
    if (is_singular() && (get_post_type() == 'my_cpt')) {
        $returnage = $content;
        $returnage .= 'Other Stuff that needs to be displayed on the single CPT';
        return $returnage;
    }

    return $content;
}

add_filter( 'the_content', 'content_my_cpt_filter', 100 );

Happy Coding,

Kuchenundkakao