Refreshing a WP site – best practice?

I’m migrating a site from one WP theme to another. The old theme doesn’t use the Visual Composer, whereas the new one does – so I’m pretty much starting from scratch on a localhost installation using the new theme, and rebuilding all the content I want to migrate on that. (It’s a company site with portfolio items on it, so it’s not huge.) This approach has the obvious disadvantage that I’ll need to clone the localhost site across to the live site once it’s finished, which doesn’t seem as straightforward a process as it should: I used All-in-One WP Migration to clone the live site to localhost, but a lot of the media library seems to have become corrupt along the way, with dead image links etc.

So, my question is – is it possible to do all this on a hidden version of the live site, and then just swap the two over when it’s done?