I hope I am not too late to answer your question. I will give you two careers path with disadvantages for both of them
Get a bachelors in either EE, ME or computer science with BME courses as your elective during junior and senior year. Then get your masters in BME. This will fulfill two purposes. First, you will have better understanding in one specific field. Second, if you can’t find a job or you live in a area where there are few BME jobs then you can alway rely on your EE, ME or computer science.
Disadvantage: none, I am serious.
OR
You can pursue BME as your bachelors and your masters. BUT… this is very important. You have to apply everywhere for getting an internship over the summer. Apply all over the nation. Get at least two internships before you start your masters. Get at least one internship during your masters. You should have total of 3 internships by the time you graduate with masters degree.
Disadvantage: BME jobs are only in selected cities. Majority of the jobs are in cold climate so if you want to move to southern states, well tough luck. Some of the top cities with BME jobs are Boston Area, Minneapolis, and California.
In recent years I have noticed that there are a lot of BME jobs are opening. But they are nowhere close enough to EE, ME or programming. I work for Medtronic and people in my team are all either ME or EE. So in other words, its easier to work for medical device company if you are NOT a BME. If I was like the chairman of all American universities, I would drop the entire BME program at bachelor level and only keep it at Masters and PhD.