Kezzer

Okay, the problem was my lack of knowledge, but this is possible to do using the following slug:

'rewrite' => array('slug' => 'what-we-do/current-projects'),

Then you must visit the Settings > Permalinks page to refresh the permalinks (this was the missing step). So, I’ve managed to get that working. This isn’t documented in any of the custom post type plugin pages that I’ve seen so far, and it’s probably something they should include.