Hello Benjamin.

Your problem seems completely different from that of Donald, that was related to the backend, You are talking about a sitemap shown inside a module in the frontend if i'm right. Also the fact that a module is visible or not relating to the domain prefix makes no sense.

The Joomla configuration.php should have the $live_site variable always blank as by default. Joomla resolves the domain dynamically, based on the domain you are using to visit the website, with www. or without www. All the components of Joomla and thus also the sitemap will reflect the current domain version.

Of course notice that it's not good that your site will have 2 domains, www and not-www. Google requires that you set a preferred domain and that will be the one correctly indexed and shown in search results. Don't carry on a double domains website to avoid penalties.