An exorcist priest does not want to give the demon any tools that he can use against the priest and the victim and a video camera, which would be used more for the curious and skeptics, would fly in the face of humility, which an exorcist priest must have.
As I recall, in the book, “Hostage to the Devil,” (PLEASE NO DEBATES) one of the documented cases involved a Jewish girl, who was possessed. However, before the family approached the exorcist priest, the girl had been under the care of a Jewish psychiatrist for several years, but she was getting worse rather than better. The family finally sought out the exorcist priest in their area, against the advice of the psychiatrist. After meeting with the girl, the priest agreed to do the exorcism, but following the usual protocol, only the priest, and one other person, usually a physically strong person, could be in the room. The psychiatrist protested and forbid that this be done without him being in the room. The priest finally gave in on the condition, that the psychiatrist must remain silent at all times. The psychiatrist agreed. When the exorcism began, the demon turned immediately on the psychiatrist and began asking him questions. The psychiatrist broke his promise to remain quiet and began talking with the demon. Once done, everything fell into the hands of the demon. The demon began bringing up personal secrets of the psychiatrist including things about his dead mother. It became so bad, the psychiatrist had a breakdown in the room, by which he had to be removed. Once done, the demon attacked the priest physically and his injuries such, that he had to spend time in the hospital. He of course knew he erred in allowing the psychiatrist into the room. After the priest recovered, he went back and did the exorcism successfully.
The point is that the priest must follow proper protocol when doing an exorcism, and a video camera, in my judgment, would violate this and could be dangerous.