Personally the investment in Backup Buddy was one of the best decisions I ever made for development and deployment of Wordpress websites. If I had just 3 plugins to choose, this would be one of them. Why? Because it makes backing up, transfer, migration of Wordpress websites just insanely easy.
What has this to do with the above issue? The posters recommending a development environment are right. With Backup Buddy you would install the plugin on the live site, run a full backup and then take it either locally or to whatever development site you want. You can be running a complete duplicate of the website in a matter of minutes at a completely different URL all links intact.
Personally I want to work locally, but my attempts have all been thwarted by an odd MAMP error. So I use a subdirectory devoted to development on our own server. I work a combination of local folders and pushing to the development server. When I'm ready I can send a client a link and they see their own website (perhaps minus a post or two but you can warn them that the "look" and "function" are kept separate to the content).
If you are developing a theme for your client there's no reason why at the end of this process you can't package your theme, upload it to the live server and switch it on knowing exactly what you are turning on.
It would solve your conflict in trying to run two menus and dual themes on the same live website.
Both sides happy.