Is it permissible to put money earned from an unlawful source of profit into the Zakat fund to be given as charity, knowing that the charity organization has many categories; so which category should be chosen to dispose of this money?

Unlawful money is not subject to Zakat (definition) and must be disposed of by spending it on the welfare of Muslims. If you give it to a trustworthy organization that collects unlawful money and delivers it to its rightful recipients, choose from their categories: disposal of unlawful money, charity, or public welfare.