Schools are required by law to provide "reasonable accommodation" and we don't have a choice as to what that accommodation is as long as "reasonable" is the key word.
To get a captionist, I would first say that an interpreter is NOT a reasonable accommodation--maybe because you don't know ASL or maybe because aliens ate all the certified interpreters. Whatever your reason is, you rule out the interpreter option. Then you make the case that captioning is the most viable reasonable accommodation.
True--if you really think that it'll benefit you, you're going to have to play "lawyer" in front of a bunch of people. You just to have to make a case for why and how they should accommodate you.