For development work, PHP runs on a Linux server located on my LAN. Periodically, I wish to upload the same project to another remote server. In the past, I just added another project with the same directory but a different remote URL and publishing account to the workgroup, but that doesn't sound like a good idea. How best to do this? Thanks

Whilst slightly inconvenient, you could just switch the project between the two FTP accounts for LAN Linux and remote Linux. Just remember to check which one is configured before transferring.

You could get your development Windows computer to directly access the LAN Linux server file area (eg as a mapped drive letter or UNC), using something like Samba on Linux. That is convenient if developing directly on Linux (I find Apache a lot faster for debugging on Linux), but doesn't resolve the scenario of development, staging and deployment using 3 different computers.