ukchris welcome to the forum.
It is essential for Clinton to go with a safe, white, Protestant, moderate male, who will be acceptable to women and non threatening to men, someone seen as capable, someone politically experienced, someone she would be comfortable with on the ticket and should she become President someone she would be comfortable working with in her administration.
She goes with Senator Evan Bayh of Indiana, who is also a former Governor of Indiana. The fact he is a sitting Senator from a state with a Republican Governor does not come into play, as Clinton will want his experience, his appeal, and his abilities.
As a correction, Clinton should go with someone with the above characteristics and qualities, and whether this individual was Protestant or Catholic it would not matter in any way, shape or form.
It just happens in this case that Bayh is Protestant, but he would still be the best running mate for Clinton whether he was Protestant or Catholic.