I doubled in biochem and bme, and I also hope to do surgery. I'm kinda in the same boat and about to start med school in the fall. Also, my parents said I should do biochem because my dad talked to a surgeon and that's what he did. Go figure.
So then what should you do?
The simple fact is that med schools don't care what you major in. A com degree in broadcast journalism will have the same weight as a physics degree with a thesis in string theory. Actually the com might be better since dr.'s are viewed as socially inept, but anyway it sucks, and so does life. So if you do BME over biochem, you're probably going to put in twice the work for no direct benefit. In fact, it might even hurt you. You'll have less time to pad your application with EC's, and your grades might suffer.
That being said, suppose you take the easier route and do biochem. You get your 3.9, your high MCAT, shadowing, clubs, etc. That's still no guarantee you'll get in to med school. My EE friend got a 3.95, 37, started a biotech company with some seed money he won at his school's business competition, and he got rejected or waitlisted EVERYWHERE he applied. Hell, I did well too and only got one acceptance. Another one of my friends with a relatively perfect app only got into one school as well.
Moral of the story
You can spend your 4 years trying to craft this perfect application as my friends and I all did, but at the end of the day it doesn't matter. You could probably throw darts blindfolded with the same odds. Major in whatever you will enjoy. If you like BME more, I'd say go for it. Or you could just do both like I did.
A possible better option would be to create your own degree. Most schools have this and are pretty liberal about it. That way you could pick all the classes that you think would benefit your goal and avoid waisting time with the rest.