I see two questions.
1. can the keyboard be used even if the Logi software is not present? To answer this question, I uninstalled the Logi software from my Windows 10 computer and restarted the computer. It is in this state that I am writing this text. The answer is therefore clear: yes.
Does the connection between the keyboard and the office computer work? I cannot answer this question. But if a smartphone can be connected to your office computer via Bluetooth, there should be no problem with the keyboard.

If you can't install anything at all, you can. The receivers remember their devices, so you can programme the receiver at home. To achieve the best performance, however, I would suggest 2 unifying receivers, then you don't have to plug them in all the time, and the keyboard is quite heavy, so I don't think it's the best idea to carry it around.