I'm sorry you are going through this, but I don't have any advice besides to emphasize how you will study next time, what you learned from the experience and how you want to improve and then begging for another chance.
ATI tests are used because they are standardized and reflect NCLEX type questions. The rationale from the instructors is that if you cannot successfully pass the ATI in this course then it's unlikely you would pass that section on the NCLEX. It's a valid point. If you did not master the material and cannot pass the exam with the minimum requirement then you should retake the course to get a better understanding of the material. Is it a retake of the exact same exam? If so, I can see why they would not want you to retake it again. At that point, it's more about you remembering the answers to individual questions than it is about you actually knowing the material well enough to pass.
There is no "one question exemption" on the NCLEX. If you are one question below passing, then you fail. If you are one question below passing in one section of the NCLEX then you fail. No appeals, no exemptions. You repay the fee and retake the test. It may be beneficial for you in the long run to retake the course and use the year to study the rest of your nursing material. It would be too bad for you to be allowed to take the ATI again but end up failing the NCLEX because you don't have the mastery.
In regard to transferring your credits- it's unlikely. Specific nursing courses and clinicals do not usually transfer between schools
because each curriculum is different. Most require that you take the majority (about your last 60) of nursing credits at the school. So if you transfer you won't be able to graduate for another two years at least.