Even though it's true that the video would only be available from the Berkman Center's website at the time of the hearing, what the RIAA fails to mention is that it will subsequently be available for anyone to use in part or in full. The RIAA could even use it themselves.
It's an odd position given the RIAA's previous statements that the P2P lawsuits are intended to call attention to their rights and remedies as copyright holders rather than simply punishing individuals. And yet when given the chance to reach the exact demographic they're targetting it becomes a problem.
It looks like the real issue isn't about the video's availability, but rather their lack of control over the coverage. Maybe they are just afraid people will get a chance to see them lose yet another case and decide to defend their rights more vigorously.