Are there any tools for automatically moving code from testing to (stage and then to) production?
We are running on a LAMP stack and don't want to rely on FTP access?
Other than that, is there some BASH or PERL script out there for this same process?
Our production servers cannot run SVN or HG which we use for development.
Check out PHING. It's very flexible and you can extend it with raw php code if you need to... It does (and can do) a whole lot more than just moving things around (including testing, building, documenting, etc)...
As for the file-transfering bit, you can connect to the remote server using a bunch of built in methods (including but not limited to FTP, SCP, SFTP, RSYNC, and more)...