MacOS or Win or both?
On MacOS use scp (see 'man scp') in Terminal. I'm not sure wether it is included in MacOSX or if I got with Apple CLI Tools.
Call it via "calculated AppleScript -> do shell script ...
There is a command line utility - ftp, both on a Mac and Windows. On a mac you can also try curl. All could be triggered from FileMaker with perform AppleScript or Send Event
On the other hand, you probably do not need to use FileMaker at all. It sounds very similar to some back up procedures, so you might just run a bash/batch script or research on a backup software which can save to FTP and rename files.
csp is included as standard in OSX. Isn't being used for back ups?
Do you use MBS Plugin?
it can do ftp upload, but I would always prefer sftp for more security.
I'm looking for a utility that can be installed at a client site. No programming involved. Doesn't have to be written in Filemaker. Needs to work on both Mac and Windows.
I think Nicolai is on the right track suggesting that backup software might be the solution.