Now the customer believes that the debils are working on the project, wants to replace the timlid and accelerated control, the timlid in response to this threatens to resign and remove half-team.

Both of them make a stake.
You are forced to repeat what they have said to each other (the customer is divorced in honesty for one or two, the lead is just awful enough to sweep).
Lida is depressed for hatred.
You calmly tell the customer that his wishes are either impossible, explaining the reasons in detail, or calling the price for them. Everything is discussed with lead, carefully and correctly. Psyche - scratch the article on the spot.
Then you call the price for replacing the lead and control.
Yes, the customer is always right, but never right for free. Any additional cane costs money. For a couple of billions, we can swim naked. If he thinks that he has made an honor by placing an order, you smile and say that the chip in general is that the order means money. Excess effort is new money. We do not work for free, we do not do double amount for one price.
If the hoodies are accepted, you pay a double premium instead of the selected one for not being twice off the box. You show facts and say you could get a triple if you were more polite.
If the replacement of the lead is accepted, the old one gets a little extra for extra labor (he is dealing with it), the rest is for work.
You want to pay, or you don’t want to pay. All is simple. As a moral compensation for one stunt, the entire office will not be splashed out. He will be punished, that’s all.