There are several misunderstandings here. First of all, the US exclusionary rule applies only to evidence gained by the police, or by people acting as agents of the government, and not always to them. Secondly it applies only in criminal cases.
The question does not say which state this would be in, and these are largely matters of state law, so it makes a difference. But I don't know of any state where taking a video without explicit consent, in a place where the person has a right to be, is a crime. In some states it would not even be a tort.
If a video is taken without the subject's consent, that may be an invasion of privacy, and the subject might be able to sue (not "file charges). In such a case the video itself would absolutely be put in evidence, and if it recorded verbal permission to take the video, the case would be promptly dismissed, quite possibly with sanctions for a frivolous lawsuit.
Even if the video were taken by a police officer, and was presented as evidence in a criminal case, the office could testify to the verbal permission. That would be enough for the judge to view the video as part of a suppression hearing (which is not before a jury). If the judge saw and heard verbal permission to take the video, that would be an end to the motion to suppress, unless it was claimed that the permission was somehow coerced or faked, and evidence supported such a claim.