The Cloth Pinning vertex group is what determines how much the mesh acts like cloth during the sim. Vertex weight 1.0 means it does not react at all like cloth, and is free to be influenced by other means like an armature. Weight zero mean it acts entirely like cloth and will accept no other influences. Intermediate weights allow multiple influences, tricky to balance, but do-able.
In your situation, the fully-bound end of the cloth and that which moves via the armature should have weight 1.0, probably fading to zero over an area. Then weight the “mobile” part to the bone that will move it – these weights are kept separate from Pinning in the way Blender deals with them. Just don’t name a bone with the Pinning Group’s name!