Show custom menus in seven theme
I am afraid that Seven's page.tpl.php does not render these menus. You will need to subtheme it and modify tpl.php files to render the menus. I know you stated that you want to avoid this; however, you really should do this anyway because if you are using the theme in your app then you will at some point likely want to modify it in some way, and you don't want to customize a core theme because it will break on updates, etc.

It's worth it, because Seven is actually a great theme for internal web apps (I use it for this). 

Seven's navigation was meant to be provided by Drupal's tab system. I suggest rendering your main menu and secondary menu as tabs to leverage the navigational design of Seven.