Is it permissible to demand one's private money and pressure one's father to take it from him, or is that considered an act of disobedience?

As long as your father is not in need, you are not obligated to give him anything from your money except as a voluntary act, and this is an act of perfect righteousness and benevolence. Your father has no authority over your money as long as you are mature and of sound judgment, and you have the right to claim it without committing an-aqq (disobedience to parents), while still being obligated to treat him with righteousness and benevolence.