Knuffels can absolutely be traded!
When you start a trade, there's an option on the left, near where you would enter

and

that says "knuffel".
If you click it, then it switches from your wardrobe to your knuffel.
And thank heavens for the "search" function because it shows ALL your knuffel, including the hidden ones, and if there's any sort of order to how they show up, I've not found it yet.
So no worries, we don't have to try for the coordinated knuffel-drop.
Thank goodness, because that doesn't work for things like event knuffels (it just goes straight to the

costing section) and since I don't see that knuffel at Knufftropolis at all, it might just make them disappear for good.