I'm sorry to say, but you have no chance with a BTEC - to put it frankly, it is pretty much mocked that a BTEC is actually equivalent to and worth AAA at A-levels, when its simply not the case, and hence why so many of the top universities here in the UK don't even respect them.
Its the same with top US universities too (Harvard being the toughest university in the world to get into by the by) - these US universities are infinitely more competitive to get into than even the likes of Oxbridge etc and require top grades (and when I say top, I don't mean this generically, I do mean they have to be steller). A BTEC just won't be good enough on the academic front I'm sorry to say (regardless of whatever you get on the SAT, as the SAT is not the academic curriculum you are studying at school) and irregardless of what your EC's maybe. You need a minimum base in your academics, and that has to be in the form of A-levels/IB/Cambridge Pre-U exams etc.
Your best bet would be to apply to lower-ranked schools in the US (those who offer sports scholarships, as Harvard and the top Ivy's don't) if your at the top of a sport performance wise, and look to get in that way.
Best of luck.
= Yale Student (Class of '15)