Voice recorders and storage are very cheap nowadays, keep them on you, running at all times if you ever find yourself in any similar circumstances. That way, it's no longer of "he said, she said", and it's clear who the problem is.
Of course assuming this is legal to do so where you live.
How do you envision that working? All I can think of that could in theory be possible would be using crypto to prove a video existed at a particular point in time. But that's not that valuable in terms of proof. Plus you can get the same effect by just handing a copy to someone like a lawyer who has a reputation for honesty and has a lot to lose if they're found to be dishonest.
But there's still the issue of proving that you haven't manipulated it before handing it in, and here I can't seem to think of anything better than the recording and hash submission being done by a DRM'd tool out of the individual's full control.