Not even after they carry her out and the people go outside with her 'blood' on their hands for the the cameras and crowd?
It looks to me they wanted to get people riled-up; possibly incite the crowd to violence.
Sorry, no. If Babbitt really was shot in the neck, there could easily have been plenty of blood sprayed from damaged carotid arteries. Some say that she was shot in the chest, which could also have produced plenty of blood. In that very crowded space behind the doors, her blood might easily have been smeared onto many people. No fake blood would have been necessary to attract the interest of reporters.
This bothers me: I tried to track down details of Babbitt's funeral, and failed.