This can be done by creating the Custom Field at the Work Item level, not specific to Project Item Level so that the field can be used at both the Project and Task level. The field would contain a formula that pulls the Due Date field ($DueDate) from the Task.
The complicated part is how to pinpoint which Task to pull it from as there are likely multiple Tasks. Are you expecting this to be dynamic or for a specific task on the project. This is the part that needs to be figured out based on your needs.
Also, I am going to repost this question to the Customization section of the forum to get more eyes on this. Also, we have Office Hours - Advanced Webinars that are always available where you can ask such questions and get responses from our Experts with Visual answers. Here is the link to register...