How can I safely move a module to another folder? I created a module through Drupal Console, unfortunately it wasn't placed in a folder of its own, rather straight into /modules/custom. Before I noticed where it had gone, I enabled the module. 

In Drupal 6/7, this was easy to fix - I could literally just put the files into the correct path, clear the cache (maybe rebuild the registry, it wasn't an exact science), and all was well.

The same isn't true for Drupal 8; I moved the files into a subfolder, rebuilt cache, and all requests tell me that Drupal is failing to include a required file. The particular file is a block class, and the path Drupal is trying to find it is where it used to be, not where I've moved it to.

The bit of code objecting is in Symfony's ApcClassLoader, so I guess I need to clear something that's slightly above Drupal's head to get this path reference updated.

How can I do that?

Uninstalling and re-installing the module does work, but once there's important data stored that's not going to be an option.

I was wrong, uninstalling and re-installing is not a viable option. After uninstalling and reinstalling, Drupal can no longer pick up the custom blocks the module defines. I'm guessing that reference is cached somewhere that isn't cleared when the module is uninstalled. Just need to find out where that is and how to clear it...
Just restart apache. The APC cache will be rebuild then.

You can also try to hit rebuild.php or run drush cr. Those all attempt to explicitly clear the APC cache.