Is it obligatory for the deceased's female cousin to compel the deceased's children to accept the price of the shroud that the deceased willed, or is it permissible for her to spend it for the sake of Allah?

The expenses for preparing and shrouding the deceased are taken from their estate. If they did not leave any wealth, then the expenses fall upon those who were obligated to support them during their lifetime. Otherwise, it is covered by the Muslim treasury (Bayt al-Mal), or it becomes a communal obligation (fard kifaya) upon the Muslims. Therefore, the children of the deceased woman are the first ones responsible for her shrouding. What her cousin's daughter brought for her is considered a gift, and the recipient is not obligated to accept it.