Definitely. An alternative to naming them, would be describing them. I’m inventing one right now, as an example:
“x with dot above and horizontal line below”.
There’s nothing more sighted about this than the usual alphabet that we all know. The only difference is that there are much more letters; therefore, we need someone patient enough to go through all of them and give them either names or unique descriptions. Our screen readers either go through those symbols by being totally silent, or they read only the base-letter; not very helpful.
This might be difficult to understand for someone who never encountered IPA before, but it should be much easier to implement than, for example, music notation. I would never expect a screen reader to be able to read a musical score to me, although I do admit I would be more than happy to. On the other hand, IPA represents speech, like any other world alphabet. Therefore, I still can’t see where the big problem is.
A very good day to you all!