Opening your sitemap, I notice several things that might be causing issues:
1. Your site is big, there might be PHP memory problems (do you get any info in the error log?)
2. When I access the sitemap for the first time, I get a timeout but on the second attempt, I do get a response: the source code is visible and complete but the stylesheet responds with a 404. The missing stylesheet causes the browser window to remain empty.
3. In the sitemap source, I see the W3 Total Cache notice stating you are running the cache with minification, page-on-disk/db/object caching and CDN enabled.
The fact that your site is big and the first sitemap request causes a timeout might well indicate issues with PHP memory. The plugin is designed to attempt a temporary increase at runtime of the available PHP memory for the sitemap generation process but if this is not possible (it depends on you server setup if this is allowed) and if the maximum PHP memory is too low, there is nothing the plugin can do about it. In that case, you will have to change server settings.
On the other hand, the fact that the source is finally and completely cached by W3 Total Cache indicates the sitemap generation does not run into any memory issues...
The fact that the stylesheet is not loaded suggest you are running some security plugin and/or a rule in your .htaccess (or nginx equivalent) that blocks direct access to PHP files in the plugin directory. Is that correct?
The only course of action I can suggest to quickly get to the root of the issue is to:
1. look in the server error log for related PHP memory errors
2. disable all other plugins (you might want to go into maintenance mode with one of the maintenance mode plugins for this) or at least W3TC and any security plugins and see if that gets the sitemap running properly.
3. check your .htaccess for rules that might block access to PHP files inside the wp-content directory
Let me know if you find anything...