I don't want to be a jerk, either, but I want to make a couple more points now that I have looked at the RIAA brief.
First, the proper writ is mandamus, not prohibition. See, e.g., Haas v. Kent, 803 F.2d 744, 744 (1st Cir 1986). Prohibition is about jurisdiction. I think they labeled it "petition for mandamus or prohibition" because under 1st Cir precedent the two writs are treated as interchangeable; throughout the body of their brief they ask for mandamus only.
Second, they have not abandoned their appeal. They are employing a two-track strategy. Track one is to seek an appeal and hope the district court order will be stayed while that appeal is considered. Track two is to seek an extraordinary writ to make sure this pops up on the 1st Cir's radar before Jan 22 (eight days is not a lot of time to even get an appeal docketed). In their brief, they ask for the writ or in the alternative expedited consideration of their appeal under the collateral order doctrine.