As of this morning, all outgoing calls placed from phones connected to the university PBX will use Calling Line Identity. Previously, call recipients would only see "private number" when receiving a call from PBX; now they will see the phone number of the PBX extension which is calling them.
This also means that, if the recipient has a contact for that number, or the number has been entered into the database of an application such as Truecaller, the recipient will see the name associated with the caller.