I do know that you require a medical test or prescription to get more time on the speaking test. I didn't have time to do that but somehow got 20 in my TOEFL. But I feel if you have the recommendation by the professor, you can directly go meet the concerned people or the main administration office and inform them that you have a stuttering issue and get very nervous during speaking tests that have a low time window which makes the stammering even worse and couldn't get the required score. You can just try getting an exemption for this year saying that you really don't want to miss out on being a TA this year and that you are confident of taking classes well. It never hurts to go and approach them directly and ask for an exemption considering that you are anyways not going to get it this term.