According to this, it is not that you are only “allowed” to go part time, rather if you have the faculties to carry a full credit load, your case will be reevaluated as to whether you should remain eligible.
https://www.collegeraptor.com/paying...lege-students/
Also, when you speak of the “determination” to become Greek, it kind of seems to me that you are missing the point of what fraternities and sororities are about. It’s not a reward that you get for hard work or by beating out anyone else. It is a bond of friendship.