Okay so, figured out the cause.
Our multisite uses a domain mapper plugin. It creates a mapping between mysite.com and mymultisite.com/mysite-com. Which apparently your plugin doesn't pick up on.
So basically.. all the images on the site link to things like mysite.com/files/2016/03/myimage.jpg, but the multisite version also works.
So, unless you have some code I can put into functions.php to help it along, our only options would be to either find a different plugin, or to pull the site out of our multisite entirely and put it in its own wordpress.
We have many many sites in this multisite and don't have individual site FTP set up currently, so I don't feel comfortable giving you FTP access.