It seems that Confluence Cloud removes the ability to edit space custom HTML and there’s no option to remove the top navigation bar because Atlassian’s says it removes the ability for users to log in (which is exactly what we want).
I do not have a solution from the top of my heart. But I’ll ask @Cbork to do some research about this for you. I also know, that my colleague Jan Kuntscher has done some research on theming in Confluence Cloud. This is limited and the freedom for apps is much more limited than in Server or Datacenter.