I've been bipolar for 30+ years and never lost a job because of it. I was also highly successful in the nnational journalism field and in law. I even went to law school at Georgetown at night and worked by day, which was a feat! It's do-able as long as you find medications to help stabilize you and motivate you. Know what pulls you out of depression (I take dexedrine routinely, have for 22 years), keeps you out of depression (Me-Effexor), and watch for signs that you're getting manic (I promise people things that I know I can't deliver on.) Learn enough to know that you have a lifelong, serious illness, and should never be without treatment and never go off your medications. Judge well how stress affects you. Medical school would be a real stressor, but if you really want to be there, it's exciting.
There are probably no reliable statistics on how many psychiatrists or therapists have bipolar disorder. (Loads of them suffer at least from occasional depression, but many lie about it.) My father became a psychiatrist but only self-medicated and died of suicide. Today, insurers may give you a hard time, or occasionally check your judgment on cases, but your peers will be fine. You may even find that professionals will go to you because you have bipolar disorder.
I had a peer counselor who had bipolar. Frank was a great motivator and counselor, but wound up in the hospital about every two weeks. Mentally ill people need someone who will be there for the long haul.
If you become a psychiatrist, be on the cutting edge and know about cardiovascular disease treatment for those who have been on antidepressants for decades.
I don't think you will find anything that will give you odds. It will all be anecdotal. Just get a good psychiatrist, and good therapist, and get as stable as you can before applying to schools. They will ask at some point or another whether you have mental illness. I had to tell the North Carolina State Bar, and had to attend a hearing on whether I should be allowed to take the bar exam in NC. I went with letters from former employers, and a time line showing that at every single moment after being diagnosed that I always had a psychiatrist and most of the time a therapist. I also showed that I had told one lawyer in my community who agreed to handle my cases should I be hospitalized. It worked. They didn't even have one question.
Good luck to you. You may have an extra hurdle or two to go through in getting there, but after that you'll be fine.