Maybe you can modify the FME Server Engine resource to point to a common share used by all your FME Server and make sure this share is always updated with the latest version of the code.
What we have internally here is a custom tool monitoring source control check-ins (Subversion) and taking care of redeploying the code (workspaces, custom transformers, python file and any configuration change (role, users, notification, schedule)) as soon as there is a change.
Thanks @larry. Unfortunately I don't have the permissions to make that happen. I could potentially set up a workbench that crawls the various servers to monitor/push new versions of the code...!
If you keep your common scripts in a version management system such as Git or SVN, it would be fairly easy to set a scheduled task on each server to update them daily (e.g. a nightly 'git pull').
If the use case is moving solutions from Development to Staging to Production environments, then this Q&A; might be of use:
https://knowledge.safe.com/questions/35716/how-can-i-move-a-solution-from-my-development-envi.html?childToView=50984#answer-50984
In short, I'd consider placing the python scripts in a Resources directory on FME Server and then moving them and other parts of your solution with the Projects feature in FME Server (2017.0 or newer). I'd be interested in feedback on that idea.