One, it isn’t a violation of copyright law to record something that is trademarked or copyrighted in incidence to a documentary film – that would make filming in general impossible in areas where copyrighted work is prevalent “ads, etc”. Were you to use the recording as means for distributing the song itself or use the song as a way to work with your movie, it would be – but talk to the producer and holder of the copyright – they aren’t bad people and often they will set you up with licensing.

Two, while there is a case where copyrighted material in incidence to a film was attempted to force a takedown (LENZ V. UNIVERSAL MUSIC GROUP) yet even in that instance the EFF provided her with legal counsel and she wasn’t found liable for the infringing content, its still on Youtube. This is often used to commit the fallacy of exception – because this one woman got a request to remove her content that involved another’s work incidentally, obviously all copyright must be destroyed.

Three, even if the above were true, taking the ringtone out of a documentary and causing it to “lose credibility” is a shaky argument. I’d like to think documentaries are primarily thought of as credible based on how they are researched and filmed, rather than if the ringtones are all there or not.