Did you happen to find a solution to this. I am also up against this brick wall, and would appreciate any advice...
You can set up an extra web gateway with a redirect rule. For example, if your url is: http://my_company_tableau.com, you can reassign that url to a different web server and then set a rule in that server to redirect any requests to http://my_tableau_server/#/my_custom_landing_page. Since the page you're redirecting to is still behind Tableau Server's firewall you don't have to worry about any security leaks.
A bit of a hack, but it works. Tableau wouldn't support this particular solution--hopefully they'll build it into Tableau Server in the near future.