as usual just talking about the problem to better understand it helped a lot.
I just tried an idea I had this morning but forgot to try:
Just adding it like this: http://126.96.36.199:2008/
(notice the extra / at the end)
And voila! It plays fine even outside a Media Scenario, just from the regular media file browser interface!
So I will think a bit more about this to see what this means. I would like to try and import all shoutcast streams (they provide some sort of xml feed containing all streams) into the LMCE database, as long as there is a sensible way to find them afterwards. I can probably write the station name as a "title" attribute...