I’m not 100% sure, but I think COCKPIT_SITE_DIR should point to /htdocs/cms, too. I only had one Cockpit project on a strato host and I used cockpit in a subdirecotry without a sub domain. So example.com/cockpit worked with just defining
<?php
// strato environment, root is in subdir, but domain points to root/subdir
// cockpit is in root/subdir/cockpit
if (!defined('COCKPIT_BASE_URL')) define('COCKPIT_BASE_URL', '/' . basename(__DIR__));
if (!defined('COCKPIT_BASE_ROUTE')) define('COCKPIT_BASE_ROUTE', COCKPIT_BASE_URL);
if (!defined('COCKPIT_DOCS_ROOT')) define('COCKPIT_DOCS_ROOT', dirname(__DIR__));
Maybe you have to set RewriteBase / in .htaccess, too.
See also this thread
Oh, and if you are forced to use strato, don’t forget to add a .htaccess file in your public root directory that denies all or redirects to your main domain. Otherwise customer_id.swh.strato-hosting.eu exposes all files above /htdocs - and you made your customer id public in your screenshot.
Cockpit should work fine without https, but in general it is a big issue 