Overriding system.site.uuid with an environment variable
In the end, it was a matter of getting Drush to work on the remote environment.

I was also able to build a small module to update the site UUID of an existing site, however we didn't end up using it. The code for that is available at this repository.

Essentially, this boils down to running something like this:

\Drupal::configFactory()
    -&gt;getEditable('system.site')
    -&gt;set('uuid', $uuid)
    -&gt;save(TRUE);


Where $uuid is whatever the UUID you want, whether this is generated manually or loaded from a service.