fwiw, Duke is the source of the “Harvard of the South” moniker, and they have been pushing it relentlessly for years!
Your paths are strikingly different, and the years after undergrad are also different. BME for undergrad needs to be started pretty much from the beginning, b/c there are so many requirements.
If you go PhD for BME you will need a good GPA, but it is looked at more in context, and your research work during college will be an important variable. Your PhD will also be fully funded (ie, no cost to you, and a healthy stipend to live on)
If you go PhD for public policy at a high end place (eg Woodrow Wilson, Kennedy, etc.) you will also be fully funded (though the stipends are typically a little less generous than in engineering XD), but they typically want relevant work experience- usually several years- before you apply. That work experience is typically very badly paid, because the employers tend to be non-profits- and there are a lot of people who want the experience. If you have a lot of debt that may be a difficult option, because the salaries are typically just about enough to live on, with not much left over for servicing your debt.
If you go med school, you need high grades, and BME is not a grade-friendly course. No, med schools will not give you a break for the fancy name, and most likely not for having a tough major, either. Med school is, obviously, very expensive, so if that is a serious possibility take the path that will leave you the least debt.
Both your choices are super choices, and you can get anywhere you want to go from either place- but debt will follow you, and shape your choices for at least a decade (probably more) to come. It can be hard to imagine what that will really be like from where you are now- but pretty much everybody on CC over 30 knows somebody who has taken jobs they don’t prefer, nursed their old car for years, lived with parents/friends/roommates, etc. into their 30s, because they need the money to be able to live and pay of student loans. Even with good medical / law / other degrees.