I don't see why your DS would really question it to be honest. It's his mums name, it happens a lot.
So you let him stay in the house with you? I think he needs to leave, while he's there you are going to be constantly on edge, and he will not let up on the pressure.
If you intend to let him stay I would make a deal. He can stay in the house to be with his son but you get the naming rights. Maybe put a time limit on it. If he wants his last name used then he has to move out.