Is zakat obligatory on money designated for a charitable project, knowing that it has reached the nisab and a full hawl has passed over it, and that there are fatwas stating that zakat is not obligatory on money designated for such projects?

If you remove the amount from your possession and it becomes an endowment (waqf) for the project, then zakat is not obligatory on it, even if a hawl (a lunar year) passes over it and it reaches the nisab (minimum threshold), because it has become the property of the charitable project, and not personal property.