Is it permissible for the father and brother to take the money saved by the girl from her work, and would she be considered sinful if she does not give it to them, especially since she is unwilling and needs the money at the beginning of her married life?

What your father and brother claim, that it is impermissible for a woman to leave her father's house with money, is baseless. The money is yours, and you may dispose of it as you wish. You are not obligated to give it to anyone if you are in need of it. You do not sin by withholding this money from your father while you are in need of it and he is not. Your father is not obligated to provide for your dowry or trousseau. If he wants the money to prepare your trousseau, then you should give him what is sufficient for that, and whatever remains is yours.