A very neat puzzle! I actually can't decide which one is correct.
One thing I thought of was the idea of quantum superposition: sleeping bauty has not observed the coin toss, so(from her point of view) it is simultaneously heads and tails, so all the three "wakings" are equally real.
Given that, I think if sleeping beauty were an electron, she should choose 1/3 because only 1/3 of the equally-real "waking events" invilve a heads coin.
But sleeping beauty is not an electron, and the false branch is not real whatsoever, so the analogy kind of breaks down.