Sorry for opening a second threat but I think I started at with the wrong plugin: https://premium.wpmudev.org/forums/topic/urgent-both-the-subdomain-and-the-mapped-domain-are-showing#post-1177028
Now after some experiments we are sure, that the domain mapping plugin is causing a 301 redirect from HTTPS to HTTP for the main domain of our multisite network.
When we used the Pro Sites plugin in addition, everything worked with HTTPS. Without it, we cant't get HTTPS to work due to a 301 redirect.
When we force HTTPS within the domain mapping plugin (or Plesk) then the website is not reachable.
What can we do?