I think it's more about giving Barr the finger.
Berman didn't like the way Barr handled it since Barr doesn't really have the authority to fire him, and he also wanted to keep his people in the post.
I don't know that anyone had a problem with Clayton, He just got caught in the middle and once Lindsey said he would honor blue slips Barr figured he wouldn't be able to get Clayton confirmed anyway.
Who told you that the Attorney General doesn’t have the authority to fire Berman? He just proved he does. If the president refused to back his AG in this, the AG would have no recourse but to resign immediately. Not supporting the Attorney General would demonstrate the President had no confidence in the AG, and would completely undermine his authority to run the Department of Justice.
In an ordinary business situation, the President can delegate this authority to fire anyone in the executive branch to Barr, or whomever he wishes. However, I suspect in this case, that Barr chose to involve the President in order to make the slap sting more soundly.
Barr- Mr. President, I want to fire so-and-so. He has been one of your persecutors and really up to so knarly shit like for instance not turning over evidence to Durham and otherwise covering up for his predecessors...
Well, Bill, fire him.
That's a verbal order from the President. Barr then executes the order.
People keep thinking that it isn't an order unless it is in writing and signed by the person who has the authority to give the order. That just isn't true.