Luckily I've never had to do a big poison ivy removal like this, but I have done some other vine removal, just by pulling it up as much as possible every time it comes up, and trying to pull up as much roots as possible every time.
I AM very sensitive to poison ivy, and so was my father, and we have always used a method that is very effective for preventing the rash, if you know you might have been exposed. I've even pulled up the vines intentionally with bare hands, and then prevented the rash. The rash is caused by an oil called urushiol that gets from the plant on to your skin, and then over the course of several hours goes deeper into your skin and causes the rash, which as I'm sure you know, only shows up a few days later. The roots and dormant winter stems also can cause a horrendous rash too (ask me how I know). But if you wash the oil off very thoroughly with soap or detergent after you think you might have been exposed, you won't get the rash. You also have to think carefully about any ways you may have spread the oil onto something else that you might touch later, such as your shoes, or the insides of the sleeves of a jacket (ask me how I know that one), and make sure to wash everything that might be carrying the oil. You don't need special detergent. I've used whatever soap or detergent was around, and it works, but I'm very reactive and have some horrendous childhood memories of poison ivy, so I usually lather up and rinse down twice, and make sure to remove any relevant clothing directly into the washing machine. That always prevents the rash for me. I've even gone back to wipe down a doorknob with soap. When I do get the rash, it is always when I didn't know I was exposed and so I didn't wash in time. I don't know how many hours exactly is the cut-off time, but I always tried to do it within a couple of hours and it has seemed to work.
So you could do a massive cut-down-dig-up once, and then every time it resprouts, pull up all that you can find. Be sure to do it when you are wearing washable shoes or rubber sandals or something, and head straight to the washing machine and sink or shower afterwards.
Covering the whole area with a very light-proof tarp for the whole season might do the trick with hands off -- it does kill most plants, except a few that have vigorous root systems and can traverse and come out of the sides, out into the sun.