Thank you Michael and Sebastian.
I have the server virtualized on my desktop and like to make snapshots after each step so I can easily roll back if something goes wrong, or move back to the initial state to test my step-by-step instructions before rolling them out.
Unfortunately the changes are not just the layout. A lot of the content will change and the new theme has different panes and such. We also move from a design that shows multiple pages connected with "Next" and "Previous" buttons to a single page layouts where users will have to scroll a lot. Thanks to Facebook, Twitter and co they are used to that and love it.
The new theme uses bootstrap and there a lot of nice things included like FAQs, carousels, tabs and so on and we want to make use of those.
I could try to port it as good as possible, only switching the theme for each page and adjust for the panes. The site will look bad during that period. Then I could work on the live site to make it look good again and use the included theme features, which may take multiple days.
I just thought there maybe some way to prevent this transition period and directly move from old to new.
Edit: So, even if I make the changes on an identical copy of the productive site, both with the same modules and themes installed, I will get into some problems with Export/Import?