Andy Burnham would be the safe pair of hands. Smooth, bland, able to keep the organisation running smoothly without having any radical impact.
Yvette Cooper would struggle because who would dare appoint another female bishop who's married to a clergyman?
Liz Kendall would be "too young", "too lightweight*", "not enough experience in parish work".
Jeremy Corbyn would be the bloke who's done 40 solid years of parish ministry, worked hard, loved by his parish, but never become archdeacon or even rural dean. Which seems really odd. Until you hear his theories on Biblical inerrancy and the Last Times.
* about 8 stone, according to the Daily Mail