When using Shared Libraries in Central, you have to manually include the file’s in your application’s product XML file via the file html tag with the src attribute to the file on your website, absolute.
I must have glazed over this in the documentation as I was wondering why my Shared SWF’s weren’t getting installed.
…however, this still doesn’t explain why they don’t work in my app. I tried changing the export frame, changing the swf path to the website path instead of locally, and a few other things, but still no dice. Maybe I’ll figure it out tomorrow.