I can understand the sentiment and could see a situation where you might be able to conduct it in a way to remove the most notable conflicts of interest (i.e. embargo on any detailed info about the documentary till after the audit, clearly stated rules agreed before the filming that are followed throughout, etc).
However, if the audit’s intent is to increase credibility, you want to be extra careful about avoiding even the appearance of conflicts of interest, which a documentary crew can still present (even if in a minor way).
More generally, transparency isn’t putting a camera crew or livestreaming your audit. Its about having clear and consistent rules that, if you gave them to another group/firm to conduct the audit, they would reach the same repeatable result.