We got in touch with ProPhoto’s excellent support team (as well as WP Engine’s) and their guys there implemented a fix quickly. It’s a known issue apparently with WP Engine’s hosting set up, but I spent a few hours investigating and trying to fix it. I couldn’t find any other posts talking about it so hopefully this’ll save someone some time.
Just get in touch with the ProPhoto team and ask them to patch your install! They say they’re going to integrate the patch into a future release so hopefully this is a short term problem.