It turns out my problem was actually a different one, but Antid Oto's post suggested the solution. I get an error message saying the drive is busy, but unmounting it does the trick.
I can't recall this coming up with Brasero in other distros, though that may be because I've never happened to try copying a disc that was mounted. It seems odd that this should be a problem for Brasero - and anyway, why doesn't it just automatically unmount the drive? and counterintuitive, because you normally mount media to get at the data, or even see it's there ... This is the sort of thing that frustrates the inexperienced and makes them go back to the Devil they Know.