Based on the OP's posts, it seems we're dealing with a TB org. IMO, when there are competitive tryouts and players are paying more than just registration fees for the privilege to play for an org/team, there's a big difference in the roles and goals of the head of a TB org v. a league commish. A TB org head's responsibilities absolutely include making sure that every team's coaching staff is in sync working collaboratively toward both team and organization goals.
I don't see the difference as being that big, TBH. The intensity level is certainly higher, but the goals and responsibilities are the same. The conversation between team and HC needs to happen before the org head takes unilateral action. Nothing in any of the OP's posts indicates that this discussion will result in a positive outcome, granted, but it can't be known for certain until it is attempted. IMO, if the AC can't talk to the HC then they're both at fault for the dysfunction - how am I to know that the AC isn't the one stirring the pot?
As the org head I would gladly take the role of mediator for the discussion, but to have a coach threaten to take his ball and go home without even attempting to remedy the situation is ridiculous.