Apologies for the long post, but this is a very fragile situation that requires great care and thought.
You're his cousin, so first sit down with him and tell him that ultimately he will need to sit down with his pops about leaving the house. It's his responsibility as a son and a crucial step because it will assert to his father and himself that he is not the type of man to go behind anyone's back.
Starting the habit of "running away" can cost a man dearly in the long run. Today you're running from your father, tomorrow you're running from your wife and kids. Announcing your departure is a staple of manhood everywhere; it shows courage, civility, and overall mastery of the self.
Secondly, assure him that his father won't do anything reckless to the family. It's quite clear that the man's bark is too loud (if you know the proverb).
If he tries to argue with you about "Oh you don't know how my father is..." check him by telling him that if things are that bad then he has to call the authorities for the well-being of the family. It's his duty as the elder son to eliminate any and every threat to the well-being of his younger siblings (since they are the future of the family, not the mother or father). What he can't do is
nothing.
Thirdly, help him get a job (either by connects or motivation). Once he lands the job, make sure he really has a place to stay with the other cousin or he can find a place to rent cheap. Only then can he tell – not ask: TELL – his father he's bouncing.The trick is for him to have everything set up to move the night he announces his departure. Also, don't let him announce his departure in an eventful date, it's best to do it in a normal night.
For this last step he can refer to the scene in the movie "There Will Be Blood" where H.W. Plainview tells his father that he's leaving with his wife to Mexico. He can just say that he's found a place and he intends to start his life. Tell him to
thank his father for making sure that he always had a roof, food and above all an education. He can complete by stating that it is now time he create the conditions for the prosperity of his own children. Tell him to keep it short and simple, to let his father have a say after he's done and to leave to never come back.
Should all of the above be done correctly, no matter how controlling his father is he will:
1. Have no grounds to hold him back;
2. Have no grounds to chastise his son for this decision;
3. Create a healthy pattern for his siblings to follow;
4. Help his mother understand that in some ways she failed them by not speaking up, hopefully causing her to pick up her slack.