The method I wrote about above about getting rid of poison oak works for three reasons. One the Round Up, but it needs to be applied during the active growth period of the plant, and because the black plastic bag deprives the plant of any light for photosynthesis, and if the bag is secure, the stump ends up being cooked to death from the high temps in my summer garden. I've never had to go back a re-treat a plant.
You can tweak the process by soaking a cloth in the concentrated Roundup and securing it tightly around the stump you have left and then putting the bag over the whole thing.
The birds usually "plant" the seeds for PO up on the slope, so I generally take two bags up with me for the whole process. It's almost as if I sneak up on the plant, bag it, cut it, treat the stump and bag the remaining stump and haul the bush/vine part off in the first bag. This is war.
If the root system is as you described (I haven't grown it) it, is the kind of plant that if the roots are broken in any way, it signals them to push up more top growth right at that spot ... very much like the roots of the rose 'Dr. Huey' which is supposed to be almost impossible to get rid of once established. I've used the same method on good ol' Dr. H. and it works with one try.