Replay Media Catcher 2.1 can capture the stream if you have MP3 Snagger turned on (The MP3 Snagger is really a "real time MP3 recorder" - the name will be changed in the next update. It works by just looking for MP3 streams as they pass into your PC, and doesn't try to re-establish a connection.) Thanx to Bill from Applian for the hint.

Also HiDownload and NetXFer can record it if you use "QuickTimeWinInet" as a user-agent.