I use Dropbox or Google Drive to share AEP files. If you need to share the AEP file, that's the easiest option. The list of the supported file types shows up if the upload fails. It's a bit hard to read, but they are all there. Somehow AEP files were left off the list.
Are you sure your AEP file was not too big? Warp Stabilizer, Rotobrush, and a few other tools can bloat an AEP very quickly.
(it seems AEP files do appear in the list of valid file)
Anyway, I think I will do as you propose and upload my files a third-party service. Probably GitHub, and share a link pointing to this service, because I fear that I'll clean my Dropbox once, and the file will be gone for everyone :S