Obviously, Romney does not pick another Mormon as his running mate, nor does Lieberman pick another Jew as his running mate.
McKinney would not be taken seriously as a contender for the Presidency and would lead the Democrats to a devastating defeat, not only for the Presidency, but down ballot as well, for House, Senate, and Governorships.
In the scenario you describe, two Mormons, both competent, two Jews, both competent, one completely irrelevant black woman, and it makes no difference who her running mate is, here's what happens.
This map is generous to McKinney.
Romney/Huntsman 284
Lieberman/Bloomberg 235
McKinney/ Dodd 19