Fixed... and broken.
Pulled it apart for one last shot. Tried to seat down the seal with a tiny bit of grease, which helped keep it in position. Held it in place carefully and screwed it tight. Noticed that that the central section seemed to be moving a round a tiny bit, so tightened the screws a bit more - pop! One of the screws has come through the other side - literally popped a tiny alloy 'cap' from the outside of the carb.

I laugh to myself (rather than cry) and think, 'Oh well, I'll reassemble it anyway just to see what happens.'
The scooter starts second kick, idles well, revs cleanly everywhere. Take it for a spin and all seems well, but the plug shows very lean settings. Raise the needle one position and it's unhappy at 1/4 throttle.
This is the exact situation I had when I started!
Will probably try to methodically richen it and do some proper plug chops, but try to keep it rideable at 1/4. But I can't really trust the carb with the damage I've now done, so looks like some kind of replacement is in order. I need to contemplate whether I stick to what I (kind-of) know, or go to a Dellorto or Mikuni.
Cheers,
Dan