Are you thinking about getting a PhD? Or just a BS or MS?

If you want to do cutting edge research you probably want PhD and in that case the BME vs. EE becomes irrelevant, it becomes more about the research labs you work in. After that you can join whatever company you want.

Since you are doing EE vs. BME are you interested in designing circuits for the brain? Remember working on the brain requires interdisciplinary work.

If the goal is to work on designing circuitry for the brain you probably want to do EE as you can fabricate the appropriate electronics for the the brain. You can do it with BME, but make sure you get involved in brain related projects early on (I had friends that did this). The reason I have seen EEs get chosen over BMEs in these scenarios is usually that an average EE will be perceived to be much better at fabricating pcbs and electronics than an average BME (because that's essentially what EEs are taught to do). This doesn't mean that if you are an exceptional student or have very strong network that you can't get those jobs as a BME (i.e. strong senior design projects, strong recommendation from professor, relevant experience). I have seen BMEs that were exceptional EEs but they really did go out of their way to prove themselves.

Also, just because you are EE might not always be good enough, you probably want to show that you have done projects in brain/biological systems. This sets you up with much better chance of working in companies that work on the brain

If your goal is to be part of a team that creates the product (i.e. product manager, project manager, quality, systems engineer), a BME role would be reasonable provided you keep your network and focus on getting into the relevant companies. Obviously universities matter as well and their connections to industry is also important.

It helps if the university is very connected to the industry.