Well you have given away the answer. Obviously many will pick B.
However, whether one is born poor or rich doesn't really matter. It is the result of the condition one is born into that matters. That means what one does with the situation.
From the story, A seems to have a greedy motivation and B chose to give away money. The result is that A is born poor and B rich.
From the Lamrim, the 7 attributes of a high lineage of human life are nobility, beauty, resources (wealth), power, wisdom, health and long life. (There is an eighth attribute ie being born male which is not so applicable in this modern age as women has the same opportunities now). With these 7 attributes one can create the causes for whatever human life one wants. Human life being the most precious compared to the other 5 (hell, hungry ghost, animals, demi gods and gods), one can practice all the way to achieve Buddhahood.
The cause of be rich is generosity as the story correctly portrayed.
So, from this point of view, being born rich certainly has the advantages. Also, being born into a billionaire's family, we may deduce the attributes of nobility, beauty (haven't seen an ugly child from the rich, haha!), power, health ( better medical care, though not guaranteed).
So being born a human is very very precious. Being born with these attribute gives the opportunity to be reborn again and again as humans if one practice morality and prayers (the main cause of a human rebirth).
So in the final analysis, I will definite choose B, with one final wish; that is to meet the Dharma.