Short answer: You flag his behaviour with HR or management.
Be specific. Give exact examples. Ensure that you are not standing alone on this. If others are being inflicted with this behaviour, ensure that they go to management too. The more complaints, the more likely that it will be taken seriously.
Is management aware of his behaviour? Why would they promote him to "shut him up"? It seems to me that if there is a problem, then by promoting him management have condoned his behaviour and even faciliated it, now he can kick down on people he now perceives are organisationally below him. As part of your brief with management, I would mention how his "appeasing" promotion has made his behaviour worse.
True workplace bullying is insidious and must be taken seriously. If you want to have it actioned, you need to take it seriously and complain through the proper channels with evidence and if possible, support from other team members.