As others have said your coach should be advising you on arm bars and not medical matters unless he is qualified. See what the doctor says and follow thier advice, I'm sure your coach means well but it's not his specialist subject.
Just out of interest, what type of injury is it?, is it a rotation injury?, like the type of pain you get after being subbed by a kimura lock? - I have a should problem also with my left side, it's mainly when I throw jabs, it's bearable but I sure would like it to go away.