1. company is too small, there is no other opportunity for your career. If it was a large one, you might consider to transfer to othe department, but it's not.
You can not work for other people in the same company as the lady will get real mad if you do so, also get mad at the person who you will report to. Nobody dares to steal you from her. So it's a dead end.
2. Not quite familiar with your industry and your function. But i would think there are planty bio company, needs communication professional. If you look outside the box, just use the experience to build up your resume, (do make it look good) and you can work in a bigger company with more responsibilities.
3. Try to learn as much as possible from your boss. She has the job security thing. She doesnt want to train you. But try to learn the way she talks to external client, govement, try to get more contact with her contacts, etc..... Try to learn her thought process when she manage a project, etc.
That's all I could think about. it's experience. Dont give too much hope on your boss. I dont see any career for you under her roof