Ah, thank you for the explanation. Interesting.
There might be a somewhat devious way of achieving what you want, but it would be workable only if nearly all your questions use your own custom question type(s) rather than the standard built-in types.
There is a rarely used somewhat experimental feature that allows you to set a different jobe server from the default in any question or question prototype. So you could change your own question types to use your departmental jobe server, which you do in the Advanced Customisation panel by setting the Sandbox parameters field to something like {"jobeserver": "our.jobe.server"}). You could then tell IT to set the standard CodeRunner jobe server back to its default value (jobe2.cosc.canterbury.ac.nz). Then all other departments would be using that default jobe server, which would soon start complaining about over-use. All your questions (assuming you stick to using your own question types) would continue to use your departmental jobe server.
I doubt it's worth the effort, but it's an interesting thought experiment.