Hi folks,
Quick followup: I attempted to get the clutch (and later the variator) off today, and failed miserably. They're really on there.
Not to be deterred, I did what I thought was the next best thing and hosed the entire transmission compartment down with large amounts of brake cleaner (it cuts through just about any grime with frightening ease), making sure to get the clutch and variator surfaces.
Next, I applied automotive belt dressing to the drive belt to make sure that it wouldn't slip.
Sure enough, the problem has disappeared entirely. There's no longer any slipping or shuddering. Total cost: about $4.
I purchased an extra belt from the local Kymco shop, but replacing the belt would involve removing the variator, which seems to be an enormous hassle. The current belt seems to be in good shape and seems to be within its usable width -- and the manual states that one needn't even inspect the belt until 7000km (I'm at 5000km) -- so I think things should be good for the time being. I'll probably return the extra belt.