I seem to recall a 2GB soft limit on DBs, but I can’t find that in the wiki. Maybe it was mentioned in these forums. You may want to ask Support directly and I’m sure they’ll direct you to a VPS or dedi.
You have to move the data somehow, so if you don’t want to export, download, upload, and import, you’ll need to allow your new DB’s IP to access your old DB, then clone it directly. I’ve only cloned from localhost to a remote DB and vice versa, so I can’t help with the details.