Assuming that the 123456 winning must occur by legit random drawing (not a prank or a bug of some kind that is biased towards such a simple result) then I'd go for the Pope story as ]more likely to happen any given day in the present. After all, there have been historically many examples of highly ranked members of groups who sincerely defect to opposing groups, starting with St. Paul. But I confess I'm not very sure about this, and I'm too sleepy to think about the problem rigorously.

In the form you posed the question ("which is more likely to happen first") it is much more difficult to answer because I'd have to evaluate how likely are institutions such as the lottery and the Catholic Church to persist in their current form for centuries or millennia.