I'm affraid we can't speak for the BHF, but as a charity they are entitled to set whatever criteria they want to. It's important to remember that, whatever the entry requirements, a student with a first will have an advantage over a person with a 2(i), who will, in turn, have an advantage over somebody with a 2(ii) and an MSc. This doesn't mean your quest for a studentship is hopeless, but you may have to send out a few more applications than the Oxbridge graduate with a first.