If you have a Mac, and sync to it using iCloud, maybe.
When iCloud is setup on a computer, and the Data & Documents are set to sync, there will be a folder that stays in sync with iCloud; much the way DropBox does. Here is an article that coves how to find those files on a Mac:
If you can find your files there, just copy and paste them into your DropBox's Ai folder.
Other than that, all I can suggest is to contact the developer's directly. Perhaps they have a good solution. Their website is pretty good and it looks like they are responsive.