File Sync without access to router

I've got what seems to be a really tough issue. Our company needs to automate file transfers from a computer that we manage that is on a network that we don't manage.

Here's the setup:
WinXP machine on a remote network (I don't know the details of that network. It is a hospitals network so I suspect strong safety measures are in place. BTW they will not forward ports for us.)
Netdrive running on that PC is connected to our FTP server.
We need to automatically synchronize files from that PC to our FTP server.

Currently, we run logmein (since it allows for no port forwarding) and login to the remote machine to manually transfer files.