Multiple simultaneous themes drawing on same database

I am working on a WordPress site for a client, using WP primarily as a CMS. I want to be able to offer three different ‘looks’ (themes) for them to choose from, but would like to be able to give them three different links to look at simultaneously so they can compare the three different looks side-by-side.

I’ve created a draft site (in a subfolder on my own domain name) with horizontal and vertical menus, structure, categories, plugins, etc, and don’t want to reinvent the wheel each time I create a new design. I know that I can swap themes easily enough within the one subfolder to change the whole site to a new theme, but that would override the other theme and I want to be able to set up several themes for the client to peruse in their own time.

The issue I’m having is that because (as I understand it) all links in WP are absolute rather than relative, if I create a site in one subfolder, then copy the same complete site files into another subfolder and apply a new theme, the (absolute) links will point back to the original subfolder.

So my question is, can I use the same database to create three simultaneously-viewable themes, dynamically updating the same content, in effect showing the same site with three different ‘looks’ for them to choose between?

For instance, is it possible to set up e.g. mydomainname.com/clientsubfolder1 and mydomainname.com/clientsubfolder2 and mydomainname.com/clientsubfolder3, all pointing to the same database but with different themes applied to the different subfolders?

(Obviously once the client decides on which look they prefer, I’d delete the other two subfolders and make the preferred subfolder the live site).

Thanks in advance – this is my first WP site, so would appreciate any step-by-step instructions anyone is able to provide!