I read again your first post and you should do one of two things:
1. Either use the names of SOUNDS, which is what people generally use in phonetics classes. You assume that because there is the "schwa" sound, then all other sounds should have nice, simple names. Well, that is not the case.
2. Or you should turn to your phonetics teacher for clarification.
The names of characters / letters are a completely different story and no sane person would like you to learn them. I had to look into "Special characters" tab in my word processor to get the names of phonetic symbols. Again, there are characters with nice and simple names like "eth" or "theta", but many characters used as phonetic symbols just have names like "LATIN SMALL LETTER REVERSED OPEN E" and that's it.
As you are required to produce SOUNDS in your task, then it is obvious that you should stick to SOUND names and get them from Roach or Sobkowiak.