So I spent some more time horsing around with those manifest files, to no avail. I followed your advice of packing both the base and DLC1 files, but that didn't work either. (Wouldn't they get the same internal path inside the BSP anyway?)
As is often the case, the real problem turned out to be something completely different:
When using custom soundscripts, you do indeed need to pack both the script file itself and the manifest. (I took the one from DLC2 as it has the most entries, and added my stuff to the end.)
After the map starts, you then have to reset the sound system with this console command:
sv_soundemitter_flush
This can be done with a logic_auto and a point_servercommand. It causes the game to re-read the manifest file and reload all of the soundscripts. Since your map is now loaded, its custom version of game_sounds_manifest.txt is visible in the virtual file system, so it will load your custom sounds in addition to the normal ones.
I don't see this documented anywhere on the wiki, so this may be just as much of a WTF revellation to anyone as it is to me. Or, maybe this is completely old news to any of you who are used to working with the sound system, in which case can someone please tell me how hard I should be slamming my head into the desk....