If I spend 3 command tokens to switch a Haris from combat group 2 to 1, does the link break? Like, can I spend 3 token to swap them all at once, or do I have to swap one after the other, thus breaking the link?
You can't move the members of the fireteam to another combat group without destroying the fireteam due to the Fireteam Integrity rules clause: In the Active Turn and Reactive Turn, a trooper stops being part of a Fireteam in the following cases: If a Command Token or some Special Skill, Equipment, or Special Scenario rule is used to move the trooper to a different Combat Group. Anything you do to move the trooper to another combat group is going to make it leave the fireteam. It's not going to matter whether that's done by spending three tokens in sequence or somehow spending them all three at once for all three members.
