I know you said you do not want to 'hack the core files' but Joomla comes with all these great module/theme overrides that let you customise their output without loosing these changes each time you run Joomla updates. All you need is an FTP program and a text editor. It does involve customising your theme but the core files are left alone.

Absolute Menu Links
-------------------

To make all menu links output an absolute url I can show you what worked for myself.

A related question that helped point me in the right direction can be found here: [How to make a template overwrite for mod_menu with alternative layouts][1].

If I want to change how my links are output I can copy the **default_component.php** to my theme's directory and change it e.g. 

copy

**/modules/mod_menu/tmpl/default_component.php**
 
to 

**mytemplate/html/mod_menu/default_component.php**
 
It is easy to change the 3 cases at the bottom for the 3 standard links types.

From here clear your cache and test. You should now have absolute urls in your menu items.

Absolute Article Links
----------------------

Depending on the size of your site this could be easy or tedious. Ask all content editors to use absolute urls when creating/linking to content. You could use a db utility by [NoNumber][2] to search and replace in your database but always remember to take a backup first.


  [1]: https://stackoverflow.com/questions/16824795/how-to-make-a-template-overwrite-for-mod-menu-with-alternative-layouts-for-link "How to make a template overwrite for mod_menu with alternative layouts"
  [2]: http://www.nonumber.nl/extensions/dbreplacer "db replacer by NoNumber"