jefnifnerf: the reason that no one has jumped on this thread is that you are presenting a VERY TOUGH situation and no one wants to be responsible for steering you down the wrong path.
First, if you have a large amount of poison ivy in your groundcover, it is probably indigenous in your area and you will have to be on a constant vigil to get it again and again, young from seed brought by the birds - even after the initial control.
There are several possible approaches. One would be to don your neoprene clothing and boots and protective eyewear and start digging. Digging gets most of the roots, but more would come back from those that you might have missed. Obviously, if you have a huge amount of the problem weed this approach is untenable.
The second is to don your PPE (personal protective equipment) and spray each plant with a product that is "non-leaching" meaning that it will not soak into the soil after a rain and damage the good ground cover; you would have to avoid as much contact as possible with "good" plant material (yet knowing that some overspray and damage will occur).
The advantage of the above two scenarios is that you would have left the most plant material to prevent and control erosion.
A third is to eradicate the ground cover completely (a last resort, only if the poison ivy is in total control). If you did this, you must use a non-leaching product and at the same time be ready to replant a rapid growing replacement, maybe even with a thick mulch on the slopes in the hope of preventing the erosion.
Geez. This has been tough to write in that there is no good answer, I feel like the proverbial messenger - don't shoot me! Unless someone else would care to comment with a better idea...