"You'd need a perfectly recorded chain of history of the dig."
Yeah, but how do you perfectly proof in the first place, that it is really you asking those questions and not a implanted computerchip in your brain, or that you ain't living in a matrix? (Or me)
The philosophical answer is, you cannot.
But you can indeed donate to research, that they can hire more people, to check integrity in various ways, because generally, they are very low on funds. So yes, making fraud is an easy way to get reputation - but exposing fraud is as well, so personally I do trust them in this case until proofen otherwise. Because I see no indication of fraud here, do you?