The question states EVERYWHERE in the office. I do not think its a plausible solution to have a set up anywhere this user might go in the office. Even having only an additional setup in a conference room does not address the problem with the laptop.
I do think what the question is hinting at, is the user is using a docking station/port replicator at their desk. It states, the laptop does not turn on when they are in the conference room, and when at their desk, they use an external mouse, keyboard and monitor with no issue. A docking station/port replicator can supply power to the laptop negating a battery issue and support those devices. If it is a issue with the battery, they would experience it when they are not at the desk, because they are not directly plugged into power.